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Created September 9, 2023 00:19
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labeled_data.json
[
{
"post_canonical": "Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of acute angled triangle $ABC$. $A_2$ be the touching point of the incircle of triangle $AB_1C_1$ with $B_1C_1$, points $B_2$, $C_2$ be defined similarly. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ concur.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $K$, $L$, $M$, $N$ be the midpoints of sides $BC$, $CD$, $DA$, $AB$ respectively of a convex quadrilateral $ABCD$. The common points of segments $AK$, $BL$, $CM$, $DN$ divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that $ABCD$ is a parallelogram?",
"tags": [
"geometry"
]
},
{
"post_canonical": "An ellipse with focus $F$ is given. Two perpendicular lines passing through $F$ meet the ellipse at four points. The tangents to the ellipse at these points form a quadrilateral circumscribed around the ellipse. Prove that this quadrilateral is inscribed into a conic with focus $F$",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $ABCD$ be a right-angled trapezoid and $M$ be the midpoint of its greater lateral side $CD$. Circumcircles $\\omega_{1}$ and $\\omega_{2}$ of triangles $BCM$ and $AMD$ meet for the second time at point $E$. Let $ED$ meet $\\omega{1}$ at point $F$, and $FB$ meet $AD$ at point $G$. Prove that $GM$ bisects angle $BGD$.\n",
"tags": [
"geometry"
]
},
{
"post_canonical": "Two circles meeting at points $A, B$ and a point $O$ lying outside them are given. Using a compass and a ruler construct a ray with origin $O$ meeting the \ufb01rst circle at point $C$ and the second one at point $D$ in such a way that the ratio $OC : OD$ be maximal.\n",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $O, I$ be the circumcenter and the incenter of triangle $ABC$, $P$ be an arbitrary point on segment $OI$, $P_A$, $P_B$, and $P_C$ be the second common points of lines $PA$, $PB$, and $PC$ with the circumcircle of triangle $ABC$. Prove that the bisectors of angles $BP_AC$, $CP_BA$, and $AP_CB$ concur at a point lying on $OI$.\n",
"tags": [
"geometry"
]
},
{
"post_canonical": "Several circles are drawn on the plane and all points of their meeting or touching are marked. May be that each circle contains exactly four marked points and exactly four marked points lie on each circle?\n",
"tags": [
"combinatorics",
"geometry"
]
},
{
"post_canonical": "a) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+2d]$?\n\nb) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+4d]$?\n\nHere $[a, b]$ denotes the least common multiple of integers $a, b$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Monica and Bogdan are playing a game, depending on given integers $n, k$. First, Monica writes some $k$ positive numbers. Bogdan wins, if he is able to find $n$ points on the plane with the following property: for any number $m$ written by Monica, there are some two points chosen by Bogdan with distance exactly $m$ between them. Otherwise, Monica wins.\n\nDetermine who has a winning strategy depending on $n, k$.\n\n[i](Proposed by Fedir Yudin)[\/i]",
"tags": [
"combinatorics",
"geometry"
]
},
{
"post_canonical": "Find all triples $(a, b, c)$ of positive integers for which $a + (a, b) = b + (b, c) = c + (c, a)$.\n\nHere $(a, b)$ denotes the greatest common divisor of integers $a, b$.\n\n[i](Proposed by Mykhailo Shtandenko)[\/i]",
"tags": [
"number theory"
]
},
{
"post_canonical": "$2022$ points are arranged in a circle, one of which is colored in black, and others in white. In one operation, The Hedgehog can do one of the following actions:\n\n1) Choose two adjacent points of the same color and flip the color of both of them (white becomes black, black becomes white)\n\n2) Choose two points of opposite colors with exactly one point in between them, and flip the color of both of them\n\nIs it possible to achieve a configuration where each point has a color opposite to its initial color with these operations?\n\n[i](Proposed by Oleksii Masalitin)[\/i]",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Positive reals $x, y, z$ satisfy $$\\frac{xy+1}{x+1} = \\frac{yz+1}{y+1} = \\frac{zx+1}{z+1}$$\n\nDo they all have to be equal?\n\n[i](Proposed by Oleksii Masalitin)[\/i]",
"tags": [
"algebra",
"inequality"
]
},
{
"post_canonical": "Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Prove that if $\\frac{H_BC}{AC} = \\frac{H_CA}{AB}$, then the line symmetric to $BC$ with respect to line $H_BH_C$ is tangent to the circumscribed circle of triangle $H_BH_CA$.\n\n[i](Proposed by Mykhailo Bondarenko)[\/i]",
"tags": [
"geometry"
]
},
{
"post_canonical": "Find the largest $k$ for which there exists a permutation $(a_1, a_2, \\ldots, a_{2022})$ of integers from $1$ to $2022$ such that for at least $k$ distinct $i$ with $1 \\le i \\le 2022$ the number $\\frac{a_1 + a_2 + \\ldots + a_i}{1 + 2 + \\ldots + i}$ is an integer larger than $1$.\n\n[i](Proposed by Oleksii Masalitin)[\/i]",
"tags": [
"algebra",
"number theory"
]
},
{
"post_canonical": "Let $p$ be a prime, $A$ is an infinite set of integers. Prove that there is a subset $B$ of $A$ with $2p-2$ elements, such that the arithmetic mean of any pairwise distinct $p$ elements in $B$ does not belong to $A$.",
"tags": [
"combinatorics",
"number theory"
]
},
{
"post_canonical": "Let $C=\\{ z \\in \\mathbb{C} : |z|=1 \\}$ be the unit circle on the complex plane. Let $z_1, z_2, \\ldots, z_{240} \\in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions:\n(1) For any open arc $\\Gamma$ of length $\\pi$ on $C$, there are at most $200$ of $j ~(1 \\le j \\le 240)$ such that $z_j \\in \\Gamma$.\n(2) For any open arc $\\gamma$ of length $\\pi\/3$ on $C$, there are at most $120$ of $j ~(1 \\le j \\le 240)$ such that $z_j \\in \\gamma$.\n\nFind the maximum of $|z_1+z_2+\\ldots+z_{240}|$.",
"tags": [
"inequality"
]
},
{
"post_canonical": "Let $m$ be a positive integer, and $A_1, A_2, \\ldots, A_m$ (not necessarily different) be $m$ subsets of a finite set $A$. It is known that for any nonempty subset $I$ of $\\{1, 2 \\ldots, m \\}$,\n\\[ \\Big| \\bigcup_{i \\in I} A_i \\Big| \\ge |I|+1. \\]\nShow that the elements of $A$ can be colored black and white, so that each of $A_1,A_2,\\ldots,A_m$ contains both black and white elements.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Given a non-right triangle $ABC$ with $BC>AC>AB$. Two points $P_1 \\neq P_2$ on the plane satisfy that, for $i=1,2$, if $AP_i, BP_i$ and $CP_i$ intersect the circumcircle of the triangle $ABC$ at $D_i, E_i$, and $F_i$, respectively, then $D_iE_i \\perp D_iF_i$ and $D_iE_i = D_iF_i \\neq 0$. Let the line $P_1P_2$ intersects the circumcircle of $ABC$ at $Q_1$ and $Q_2$. The Simson lines of $Q_1$, $Q_2$ with respect to $ABC$ intersect at $W$.\n\nProve that $W$ lies on the nine-point circle of $ABC$.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $a_1, a_2, \\ldots, a_n$ be $n$ positive integers that are not divisible by each other, i.e. for any $i \\neq j$, $a_i$ is not divisible by $a_j$. Show that\n\\[ a_1+a_2+\\cdots+a_n \\ge 1.1n^2-2n. \\]\n\n[i]Note:[\/i] A proof of the inequality when $n$ is sufficient large will be awarded points depending on your results.",
"tags": [
"inequality",
"number theory"
]
},
{
"post_canonical": "Let $ABCD$ be a convex quadrilateral, the incenters of $\\triangle ABC$ and $\\triangle ADC$ are $I,J$, respectively. It is known that $AC,BD,IJ$ concurrent at a point $P$. The line perpendicular to $BD$ through $P$ intersects with the outer angle bisector of $\\angle BAD$ and the outer angle bisector $\\angle BCD$ at $E,F$, respectively. Show that $PE=PF$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $n$ be a positive integer, $x_1,x_2,\\ldots,x_{2n}$ be non-negative real numbers with sum $4$. Prove that there exist integer $p$ and $q$, with $0 \\le q \\le n-1$, such that\n\\[ \\sum_{i=1}^q x_{p+2i-1} \\le 1 \\mbox{ and } \\sum_{i=q+1}^{n-1} x_{p+2i} \\le 1, \\]\nwhere the indices are take modulo $2n$.\n\n[i]Note:[\/i] If $q=0$, then $\\sum_{i=1}^q x_{p+2i-1}=0$; if $q=n-1$, then $\\sum_{i=q+1}^{n-1} x_{p+2i}=0$.",
"tags": [
"algebra",
"combinatorics",
"inequality"
]
},
{
"post_canonical": "Given a positive integer $n$, let $D$ be the set of all positive divisors of $n$. The subsets $A,B$ of $D$ satisfies that for any $a \\in A$ and $b \\in B$, it holds that $a \\nmid b$ and $b \\nmid a$. Show that\n\\[ \\sqrt{|A|}+\\sqrt{|B|} \\le \\sqrt{|D|}. \\]",
"tags": [
"number theory"
]
},
{
"post_canonical": "Find all primes $p$, such that there exist positive integers $x$, $y$ which satisfy\n$$\\begin{cases}\np + 49 = 2x^2\\\\\np^2 + 49 = 2y^2\\\\\n\\end{cases}$$",
"tags": [
"algebra"
]
},
{
"post_canonical": "14 students attend the IMO training camp. Every student has at least $k$ favourite numbers. The organisers want to give each student a shirt with one of the student's favourite numbers on the back. Determine the least $k$, such that this is always possible if:\n$a)$ The students can be arranged in a circle such that every two students sitting next to one another have different numbers.\n$b)$ $7$ of the students are boys, the rest are girls, and there isn't a boy and a girl with the same number.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "A circle through the vertices $A$ and $B$ of $\\triangle ABC$ intersects segments $AC$ and $BC$ at points $P$ and $Q$ respectively. If $AQ=AC$, $\\angle BAQ=\\angle CBP$ and $AB=(\\sqrt{3}+1)PQ$, find the measures of the angles of $\\triangle ABC$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "In every cell of a table with $n$ rows and $m$ columns is written one of the letters $a$, $b$, $c$. Every two rows of the table have the same letter in at most $k\\geq 0$ positions and every two columns coincide at most $k$ positions. Find $m$, $n$, $k$ if\n\\[\\frac{2mn+6k}{3(m+n)}\\geq k+1\\]",
"tags": [
"combinatorics",
"inequality"
]
},
{
"post_canonical": "A $6 \\times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \\times 2$ subsquare of the board is [i]chesslike[\/i] if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board.\n\n[i]Proposed by Nikola Velov[\/i]",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Let $ABC$ be an acute triangle with incircle $\\omega$, incenter $I$, and $A$-excircle $\\omega_{a}$. Let $\\omega$ and $\\omega_{a}$ meet $BC$ at $X$ and $Y$, respectively. Let $Z$ be the intersection point of $AY$ and $\\omega$ which is closer to $A$. The point $H$ is the foot of the altitude from $A$. Show that $HZ$, $IY$ and $AX$ are concurrent.\n\n[i]Proposed by Nikola Velov[\/i]",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "We say that a positive integer $n$ is [i]memorable[\/i] if it has a binary representation with strictly more $1$'s than $0$'s (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$'s than $0$'s). Are there infinitely many memorable perfect squares?\n\n[i]Proposed by Nikola Velov[\/i]",
"tags": [
"algebra"
]
},
{
"post_canonical": "Let $ABC$ be an acute triangle with altitude $AD$ ($D \\in BC$). The line through $C$ parallel to $AB$ meets the perpendicular bisector of $AD$ at $G$. Show that $AC = BC$ if and only if $\\angle AGC = 90^{\\circ}$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Find all functions $f: \\mathbb{Q^+} \\rightarrow \\mathbb{Q}$ satisfying $f(x)+f(y)= \\left(f(x+y)+\\frac{1}{x+y} \\right) (1-xy+f(xy))$ for all $x, y \\in \\mathbb{Q^+}$.",
"tags": [
"algebra",
"function"
]
},
{
"post_canonical": "For a polynomial $P(x)$ with integer coefficients and a prime $p$, if there is no $n \\in \\mathbb{Z}$ such that $p|P(n)$, we say that polynomial $P$ [i]excludes[\/i] $p$. Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?",
"tags": [
"algebra",
"number theory",
"polynomial"
]
},
{
"post_canonical": "What is the minimum value of the expression $$xy+yz+zx+\\frac 1x+\\frac 2y+\\frac 5z$$ where $x, y, z$ are positive real numbers?",
"tags": [
"algebra",
"inequality"
]
},
{
"post_canonical": "Let $x,y,z$ be three positive integers with $\\gcd(x,y,z)=1$. If\n\\[x\\mid yz(x+y+z),\\]\n\\[y\\mid xz(x+y+z),\\]\n\\[z\\mid xy(x+y+z),\\]\nand\n\\[x+y+z\\mid xyz,\\]\nshow that $xyz(x+y+z)$ is a perfect square.\n\n[i]Proposed by usjl[\/i]",
"tags": [
"number theory"
]
},
{
"post_canonical": "How many positive integer sequences $(a_1,a_2,\\ldots,a_{2022})$ satisfy that $a_1<a_2<\\ldots<a_{2022}$, and \n$$a_1^2-6^2\\geq a_2^2-7^2\\geq\\ldots\\geq a_{2022}^2-2027^2$$",
"tags": [
"inequality"
]
},
{
"post_canonical": "Find all pair of primes $(p,q)$, such that $p^3+3q^3-32$ is also a prime.",
"tags": [
"number theory"
]
},
{
"post_canonical": "In an acute triangle $ABC$, $AB<AC$. The perpendicular bisector of the segment $BC$ intersects the lines $AB,AC$ at the points $D,E$ respectively. Denote the mid-point of $DE$ as $M$. Suppose the circumcircle of $\\triangle ABC$ intersects the line $AM$ at points $P$ and $A$, and $M,A,P$ are arranged in order on the line. Prove that $\\angle BPE=90^{\\circ}$. ",
"tags": [
"geometry"
]
},
{
"post_canonical": "Given an acute angle triangle $ABC$ with circumcircle $\\Gamma$ and circumcenter $O$. A point $P$ is taken on the line $BC$ but not on $[BC]$. Let $K$ be the reflection of the second intersection of the line $AP$ and $\\Gamma$ with respect to $OP$. If $M$ is the intersection of the lines $AK$ and $OP$, prove that $\\angle OMB+\\angle OMC=180^{\\circ}$.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "We are given some three element subsets of $\\{1,2, \\dots ,n\\}$ for which any two of them have at most one common element. We call a subset of $\\{1,2, \\dots ,n\\}$ [i]nice [\/i] if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element [i]nice [\/i] subset while keeping it nice, find the minimum value of $n$.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "There are $n$ sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions?\n\n[i](Proposed by Bogdan Rublov)[\/i]",
"tags": [
"combinatorics",
"geometry"
]
},
{
"post_canonical": "There is a black token in the lower-left corner of a board $m \\times n$ ($m, n \\ge 3$), and there are white tokens in the lower-right and upper-left corners of this board. Petryk and Vasyl are playing a game, with Petryk playing with a black token and Vasyl with white tokens. Petryk moves first. \n\nIn his move, a player can perform the following operation at most two times: choose any his token and move it to any adjacent by side cell, with one restriction: you can't move a token to a cell where at some point was one of the opponents' tokens. \n\nVasyl wins if at some point of the game white tokens are in the same cell. For which values of $m, n$ can Petryk prevent him from winning?\n\n[i](Proposed by Arsenii Nikolaiev)[\/i]",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Find all pairs of real numbers $(x,y)$ for which\n\\[\n\\begin{aligned}\nx^2+y^2+xy&=133 \\\\\nx+y+\\sqrt{xy}&=19\n\\end{aligned}\n\\]",
"tags": [
"algebra"
]
},
{
"post_canonical": "Let $\\triangle ABC$ be an acute-angled triangle with $AB<AC$ and let $(c)$ be its circumcircle with center $O$. Let $M$ be the midpoint of $BC$. The line $AM$ meets the circle $(c)$ again at the point $D$. The circumcircle $(c_1)$ of triangle $\\triangle MDC$ intersects the line $AC$ at the points $C$ and $I$, and the circumcircle $(c_2)$ of $\\triangle AMI$ intersects the line $AB$ at the points $A$ and $Z$. \n\nIf $N$ is the foot of the perpendicular from $B$ on $AC$, and $P$ is the second point of intersection of $ZN$ with $(c_2)$, prove that the quadrilateral with vertices the points $N, P, I$ and $M$ is a parallelogram.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $m, n$ be positive integers with $m<n$ and consider an $n\\times n$ board from which its upper left $ m\\times m$ part has been removed. An example of such board for $n=5$ and $m=2$ is shown below.\n\nDetermine for which pairs $(m, n)$ this board can be tiled with $3\\times 1$ tiles. Each tile can be positioned either horizontally or vertically so that it covers exactly three squares of the board. The tiles should not overlap and should not cover squares outside of the board.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Find all pairs of integers $(m, n)$ which satisfy the equation\n\\[(2n^2+5m-5n-mn)^2=m^3n\\]",
"tags": [
"algebra",
"number theory"
]
},
{
"post_canonical": "Pete wrote down $21$ pairwise distinct positive integers, each not greater than $1,000,000$. For every pair $(a, b)$ of numbers written down by Pete, Nick wrote the number \n$$F(a;b)=a+b -\\gcd(a;b)$$ \non his piece of paper. Prove that one of Nick\u2019s numbers differs from all of Pete\u2019s numbers.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Prove that infinitely many positive integers can be represented as $(a-1)\/b + (b-1)\/c + (c-1)\/a$, where $a$, $b$ and $c$ are pairwise distinct positive integers greater than 1.",
"tags": [
"algebra"
]
},
{
"post_canonical": "A finite set $M$ of real numbers has the following properties: $M$ has at least $4$ elements, and there exists a bijective function $f:M\\to M$, different from the identity, such that $ab\\leq f(a)f(b)$ for all $a\\neq b\\in M.$ Prove that the sum of the elements of $M$ is $0.$",
"tags": [
"algebra",
"combinatorics"
]
},
{
"post_canonical": "Let $ABCD$ be a convex quadrilateral and let $O$ be the intersection of its diagonals. Let $P,Q,R,$ and $S$ be the projections of $O$ on $AB,BC,CD,$ and $DA$ respectively. Prove that \\[2(OP+OQ+OR+OS)\\leq AB+BC+CD+DA.\\]",
"tags": [
"geometry",
"inequality"
]
},
{
"post_canonical": "Determine all functions $f:\\mathbb{R}\\to\\mathbb{R}$ such that all real numbers $x$ and $y$ satisfy \\[f(f(x)+y)=f(x^2-y)+4f(x)y.\\]",
"tags": [
"algebra",
"function"
]
},
{
"post_canonical": "In triangle $ABC$, a point $M$ is the midpoint of $AB$, and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$, and $B_1$ is the reflection of $B$ in $AI$. Let $N$ be the midpoint of $A_1B_1$. Prove that $IN > IM$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "The set $S = \\{1, 2, \\dots, 2022\\}$ is to be partitioned into $n$ disjoint subsets $S_1, S_2, \\dots, S_n$ such that for each $i \\in \\{1, 2, \\dots, n\\}$, exactly one of the following statements is true:\n\n(a) For all $x, y \\in S_i$, with $x \\neq y, \\gcd(x, y) > 1.$\n(b) For all $x, y \\in S_i$, with $x \\neq y, \\gcd(x, y) = 1.$\n\nFind the smallest value of $n$ for which this is possible.",
"tags": [
"combinatorics",
"number theory"
]
},
{
"post_canonical": "For integers $0\\le a\\le n$, let $f(n,a)$ denote the number of coefficients in the expansion of $(x+1)^a(x+2)^{n-a}$ that is divisible by $3.$ For example, $(x+1)^3(x+2)^1=x^4+5x^3+9x^2+7x+2$, so $f(4,3)=1$. For each positive integer $n$, let $F(n)$ be the minimum of $f(n,0),f(n,1),\\ldots ,f(n,n)$.\n\n(1) Prove that there exist infinitely many positive integer $n$ such that $F(n)\\ge \\frac{n-1}{3}$.\n\n(2) Prove that for any positive integer $n$, $F(n)\\le \\frac{n-1}{3}$.",
"tags": [
"algebra",
"polynomial"
]
},
{
"post_canonical": "How many arrays of non-negative integers $(a,b,c,d,e)$ satisfy $a+b+c+d+e=2022$, and these five numbers are all not three-digit numbers. \n\nRemark: $0$ is a one-digit number. ",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Find all integer values of $x$ for which the value of the expression \n\\[x^2+6x+33\\] \nis a perfect square.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $ABC$ be an acute-angled triangle, and let $D, E$ and $K$ be the midpoints of its sides $AB, AC$ and $BC$ respectively. Let $O$ be the circumcentre of triangle $ABC$, and let $M$ be the foot of the perpendicular from $A$ on the line $BC$. From the midpoint $P$ of $OM$ we draw a line parallel to $AM$, which meets the lines $DE$ and $OA$ at the points $T$ and $Z$ respectively. Prove that:\n\n(a) the triangle $DZE$ is isosceles\n(b) the area of the triangle $DZE$ is given by the formula\n\t\\[E_{DZE}=\\frac{BC\\cdot OK}{8}\\]\n",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Prove that for every natural number $k$, at least one of the integers\n\\[ 2k-1, \\quad 5k-1 \\quad \\text{and} \\quad 13k-1\\]\nis not a perfect square.\n",
"tags": [
"number theory"
]
},
{
"post_canonical": "The numbers $1, 2, 3, \\ldots , 10$ are written on the blackboard. In each step, Andrew chooses two numbers $a, b$ which are written on the blackboard such that $a\\geqslant 2b$, he erases them, and in their place writes the number $a-2b$.\n\nFind all numbers $n$, such that after a sequence of steps as above, at the end only the number $n$ will remain on the blackboard. ",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Let $\\vartriangle ABC$ be a triangle in which $\\angle ACB = 40^o$ and $\\angle BAC = 60^o$ . Let $D$ be a point inside the segment $BC$ such that $CD =\\frac{AB}{2}$ and let $M$ be the midpoint of the segment $AC$. How much is the angle $\\angle CMD$ in degrees?",
"tags": [
"geometry"
]
},
{
"post_canonical": "Prove that there exists a set $X \\subseteq \\mathbb{N}$ which contains exactly 2022 elements such that for every distinct $a, b, c \\in X$ the following equality:\n \\[ \\gcd(a^n+b^n, c) = 1 \\] is satisfied for every positive integer $n$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that\n$$(a + b + c)(ab + bc + ca) + 3\\ge 4(a + b + c).$$",
"tags": [
"algebra",
"inequality"
]
},
{
"post_canonical": "Given that $ABC$ is a triangle, points $A_i, B_i, C_i \\hspace{0.15cm} (i \\in \\{1,2,3\\})$ and $O_A, O_B, O_C$ satisfy the following criteria:\n\n a) $ABB_1A_2, BCC_1B_2, CAA_1C_2$ are rectangles not containing any interior points of the triangle $ABC$,\n b) $\\displaystyle \\frac{AB}{BB_1} = \\frac{BC}{CC_1} = \\frac{CA}{AA_1}$,\n c) $AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2$ are parallelograms, and\n d) $O_A$ is the centroid of rectangle $BCC_1B_2$, $O_B$ is the centroid of rectangle $CAA_1C_2$, and $O_C$ is the centroid of rectangle $ABB_1A_2$.\n\nProve that $A_3O_A, B_3O_B,$ and $C_3O_C$ concur at a point.\n\n[i]Proposed by Farras Mohammad Hibban Faddila[\/i]",
"tags": [
"geometry"
]
},
{
"post_canonical": "Determine all functions $f : \\mathbb{R} \\to \\mathbb{R}$ satisfying\n \\[ f(a^2) - f(b^2) \\leq (f(a)+b)(a-f(b)) \\] for all $a,b \\in \\mathbb{R}$.",
"tags": [
"algebra",
"function",
"inequality"
]
},
{
"post_canonical": "Let $A$ be a subset of $\\{1,2,\\ldots,2020\\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation:\n \\[ x^2 - xy + y^2 = n. \\]\n a) Determine $f(2022)$.\n b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Find all function $f:\\mathbb R^+ \\rightarrow \\mathbb R^+$ such that:\n\\[f\\left(\\frac{f(x)}{x}+y\\right)=1+f(y), \\quad \\forall x,y \\in \\mathbb R^+.\\]",
"tags": [
"algebra",
"function"
]
},
{
"post_canonical": "Let $ABC$ be an acute triangle, $B,C$ fixed, $A$ moves on the big arc $BC$ of $(ABC)$. Let $O$ be the circumcenter of $(ABC)$ $(B,O,C$ are not collinear, $AB \\ne AC)$, $(I)$ is the incircle of triangle $ABC$. $(I)$ tangents to $BC$ at $D$. Let $I_a$ be the $A$-excenter of triangle $ABC$. $I_aD$ cuts $OI$ at $L$. Let $E$ lies on $(I)$ such that $DE \\parallel AI$. \na) $LE$ cuts $AI$ at $F$. Prove that $AF=AI$.\nb) Let $M$ lies on the circle $(J)$ go through $I_a,B,C$ such that $I_aM \\parallel AD$. $MD$ cuts $(J)$ again at $N$. Prove that the midpoint $T$ of $MN$ lies on a fixed circle.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any $x$-meter portion of the wire, there is an $\\tfrac{x}{10}$ probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest neighbor?",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Find all real quadruples $(a,b,c,d)$ satisfying the system of equations\n$$\n\\left\\{ \\begin{array}{ll}\nab+cd = 6 \\\\\nac + bd = 3 \\\\\nad + bc = 2 \\\\\na + b + c + d = 6.\n\\end{array} \\right.\t\n$$",
"tags": [
"algebra"
]
},
{
"post_canonical": "For positive integers $a$ and $b$, if the expression $\\frac{a^2+b^2}{(a-b)^2}$ is an integer, prove that the expression $\\frac{a^3+b^3}{(a-b)^3}$ is an integer as well.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $f:\\mathbb{N}^*\\rightarrow \\mathbb{N}^*$ be a function such that $\\frac{x^3+3x^2f(y)}{x+f(y)}+\\frac{y^3+3y^2f(x)}{y+f(x)}=\\frac{(x+y)^3}{f(x+y)},~(\\forall)x,y\\in\\mathbb{N}^*.$\n$a)$ Prove that $f(1)=1.$\n$b)$ Find function $f.$",
"tags": [
"algebra",
"function"
]
},
{
"post_canonical": "Let $z_1,z_2$ and $z_3$ be complex numbers of modulus $1,$ such that $|z_i-z_j|\\geq\\sqrt{2}$ for all $i\\neq j\\in\\{1,2,3\\}.$ Prove that \\[|z_1+z_2|+|z_2+z_3|+|z_3+z_2|\\leq 3.\\][i]Mathematical Gazette[\/i]",
"tags": [
"inequality"
]
},
{
"post_canonical": "A pentagon $ABCDE$ is inscribed in a circle $\\Gamma$. And the quadrilateral $BCDE$ is a rectangle satisfying $BC=DE=1$. Suppose $AB=EA=6$. How long is the diameter of $\\Gamma$? \n",
"tags": [
"geometry"
]
},
{
"post_canonical": "In an isosceles triangle $ABC$, $AB=AC$. Take a point $P$ inside the $\\triangle ABC$ to satisfy $\\angle PAB=\\angle PBC=\\angle PCA$. If the area of $\\triangle PAB$ and $\\triangle PCA$ are $5,4$ respectively. Find the length of the segment $BC$. ",
"tags": [
"geometry"
]
},
{
"post_canonical": "Fill in one of the three letters A, B and C in each grid of a $2\\times 2$ table. How many ways are there to fill in, such that the letters in any two grids with a common side are different? \n\nRemark: It is legal that some character may not be written even once in the square. If two writing methods can be the same only after being rotated or flipped, we still treat them as two different methods. ",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "A palace has $32$ rooms and $40$ corridors. As shown in the figure below, each room is represented by a dot, and each corridor is represented by a segment connecting the rooms. Put $n$ robots in these rooms, such that there is at most one robot in each room. Each robot is assigned to a corridor connected to its room. And the following condition is satisfied: \n\n$\\bullet$ Let all robots move along the assigned corridors at the same time. They will arrive at the rooms at another ends of the corridors at the same time. During this process, any two robots will not meet each other. And each robot will arrive at a different room in the end. \n\nAssume that the maximum value of the positive integer $n$ we can take as above is $N$. How many ways to put $N$ robots in the palace and assign the corridors to them satisfying the above condition? \n\nRemark: Any two robot are not distinguished from each other. ",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "For any reals $x, y$, show the following inequality:\n\n$$\\sqrt{(x+4)^2 + (y+2)^2} + \\sqrt{(x-5)^2 + (y+4)^2} \\le \\sqrt{(x-2)^2 + (y-6)^2} + \\sqrt{(x-5)^2 + (y-6)^2} + 20$$\n\n[i](Proposed by Bogdan Rublov)[\/i]",
"tags": [
"algebra",
"inequality"
]
},
{
"post_canonical": "The sides $AB, BC, CD$ and $DA$ of quadrilateral $ABCD$ touch a circle with center $I$ at points $K, L, M$ and $N$ respectively. Let $P$ be an arbitrary point of line $AI$. Let $PK$ meet $BI$ at point $Q, QL$ meet $CI$ at point $R$, and $RM$ meet $DI$ at point $S$.\nProve that $P,N$ and $S$ are collinear.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $ABC$ be a triangle with $\\angle A=60^o$ and $T$ be a point such that $\\angle ATB=\\angle BTC=\\angle ATC$. A circle passing through $B,C$ and $T$ meets $AB$ and $AC$ for the second time at points $K$ and $L$.Prove that the distances from $K$ and $L$ to $AT$ are equal.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Find all prime numbers $a$ and $b$ such that\n$$\n20a^3-b^3=1.\n$$",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $I$ be the incenter of a non-isosceles triangle $ABC$. The line $AI$ intersects the circumcircle of the triangle $ABC$ at $A$ and $D$. Let $M$ be the middle point of the arc $BAC$. The line through the point $I$ perpendicular to $AD$ intersects $BC$ at $F$. The line $MI$ intersects the circle $BIC$ at $N$.\nProve that the line $FN$ is tangent to the circle $BIC$.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.\n",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that\n\\[ \\left| \\frac{a_i}{a_j} - \\frac {a_k}{a_l} \\right| \\le C. \\]",
"tags": [
"algebra"
]
},
{
"post_canonical": "Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \\times n$ rectangular array of positive integers such that\n[list]\n[*]each row contains $n$ distinct consecutive integers in some order,\n[*]each column contains $m$ distinct consecutive integers in some order, and\n[*]each entry is less than or equal to $s$.\n[\/list]\n\n[i]Proposed by Ankan Bhattacharya.[\/i]",
"tags": [
"algebra",
"number theory",
"polynomial"
]
},
{
"post_canonical": "Prove for all positive reals a,b,c,d:\r\n\r\n\r\n$ \\frac{a\\minus{}b}{b\\plus{}c}\\plus{}\\frac{b\\minus{}c}{c\\plus{}d}\\plus{}\\frac{c\\minus{}d}{d\\plus{}a}\\plus{}\\frac{d\\minus{}a}{a\\plus{}b} \\geq 0$",
"tags": [
"inequality"
]
},
{
"post_canonical": "Solve in the set of real numbers:\r\n\\[ 3\\left(x^2 \\plus{} y^2 \\plus{} z^2\\right) \\equal{} 1,\r\n\\]\r\n\r\n\\[ x^2y^2 \\plus{} y^2z^2 \\plus{} z^2x^2 \\equal{} xyz\\left(x \\plus{} y \\plus{} z\\right)^3.\r\n\\]",
"tags": [
"algebra",
"inequality"
]
},
{
"post_canonical": "Two equal circles $S_1$ and $S_2$ meet at two different points. The line $\\ell$ intersects $S_1$ at points $A,C$ and $S_2$ at points $B,D$ respectively (the order on $\\ell$: $A,B,C,D$) . Define circles $\\Gamma_1$ and $\\Gamma_2$ as follows: both $\\Gamma_1$ and $\\Gamma_2$ touch $S_1$ internally and $S_2$ externally, both $\\Gamma_1$ and $\\Gamma_2$ line $\\ell$, $\\Gamma_1$ and $\\Gamma_2$ lie in the different halfplanes relatively to line $\\ell$. Suppose that $\\Gamma_1$ and $\\Gamma_2$ touch each other. Prove that $AB=CD$.\n\nI. Voronovich",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Find all functions $f: R \\to R$ and $g:R \\to R$ such that $f(x-f(y))=xf(y)-yf(x)+g(x)$ for all real numbers $x,y$.\n\nI.Voronovich",
"tags": [
"algebra",
"function"
]
},
{
"post_canonical": "Prove that there exist many natural numbers n so that both roots of the quadratic equation $x^2+(2-3n^2)x+(n^2-1)^2=0$ are perfect squares.\n\nS. Kuzmich",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$,\n$K=AN \\cap BM$, $L=CM \\cap DN$. Find the smallest possible $c\\in R$ such that $S(MKNL)<c \\cdot S(ABCD)$ for any convex quadrilateral $ABCD$. \n\nI. Voronovich\n",
"tags": [
"geometry",
"inequality"
]
},
{
"post_canonical": "Let $g(n)$ be the number of all $n$-digit natural numbers each consisting only of digits $0,1,2,3$ (but not nessesarily all of them) such that the sum of no two neighbouring digits equals $2$. Determine whether $g(2010)$ and $g(2011)$ are divisible by $11$.\n\nI.Kozlov",
"tags": [
"number theory"
]
},
{
"post_canonical": "In an acute-angled triangle $ABC$, the orthocenter is $H$. $I_H$ is the incenter of $\\vartriangle BHC$. The bisector of $\\angle BAC$ intersects the perpendicular from $I_H$ to the side $BC$ at point $K$. Let $F$ be the foot of the perpendicular from $K$ to $AB$. Prove that $2KF+BC=BH +HC$\n\nA. Voidelevich",
"tags": [
"geometry"
]
},
{
"post_canonical": "Find all positive integers $n$ such that $n=q(q^2-q-1)=r(2r+1)$ for some primes $q$ and $r$.\n\nB.Gilevich",
"tags": [
"number theory"
]
},
{
"post_canonical": "For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$\r\n\r\n(a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively.\r\n\r\n(b) Show that, for any set $ V$ consisting of $ n \\geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$",
"tags": [
"combinatorics",
"geometry"
]
},
{
"post_canonical": "Let $a,b,c,d,x,y$ denote the lengths of the sides $AB, BC,CD,DA$ and the diagonals $AC,BD$ of a cyclic quadrilateral $ABCD$ respectively.\nProve that $$(\\frac{1}{a}+\\frac{1}{c})^2+(\\frac{1}{b}+\\frac{1}{d})^2 \\geq 8 ( \\frac{1}{x^2}+\\frac{1}{y^2})$$",
"tags": [
"geometry",
"inequality"
]
},
{
"post_canonical": "Given real numbers $a,b,c,d$ such that $\\sin{a}+b >\\sin{c}+d, a+\\sin{b}>c+\\sin{d}$, prove that $a+b>c+d$",
"tags": [
"algebra",
"inequality"
]
},
{
"post_canonical": "Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "a) Determine all functions $f:\\mathbb{Z}\\rightarrow\\mathbb{Z}$ such that\\[f(x-f(y))=f(f(x))-f(y)-1\\]holds for all $x,y\\in\\mathbb{Z}$. (It is [url=https:\/\/artofproblemsolving.com\/community\/c6h1268817p6621849]2015 IMO Shortlist A2 [\/url])\nb) The same question for if \\[f(x-f(y))=f(f(x))-f(y)-2\\] for all integers $x,y$",
"tags": [
"algebra",
"function"
]
},
{
"post_canonical": "Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \\cdots$ defined by \\[ a_0 = M + \\frac{1}{2} \\qquad \\textrm{and} \\qquad a_{k+1} = a_k\\lfloor a_k \\rfloor \\quad \\textrm{for} \\, k = 0, 1, 2, \\cdots \\] contains at least one integer term. ",
"tags": [
"function",
"number theory"
]
},
{
"post_canonical": "Let $v(n)$ be the order of $2$ in $n!$. Prove that for any positive integers $a$ and $m$ there exists $n$ ($n>1$) such that $v(n) \\equiv a (\\mod m)$.\n\nI have a book with Mongolian problems from this year, and this problem appeared in it. Perhaps I am terribly misinterpreting this problem, but it seems like it is wrong. Any ideas?",
"tags": [
"modular arithmetic",
"number theory"
]
},
{
"post_canonical": "How many ways to fill the board $ 4\\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Find all real numbers $x$ that can be written as $$x= \\frac{a_0}{a_1a_2..a_n}+\\frac{a_1}{a_2a_3..a_n}+\\frac{a_2}{a_3a_4..a_n}+...+\\frac{a_{n-2}}{a_{n-1}a_n}+\\frac{a_{n-1}}{a_n}$$\nwhere $n, a_1,a_2,...,a_n$ are positive integers and $1 = a_0 \\le a_1 <... < a_n$",
"tags": [
"algebra",
"number theory"
]
},
{
"post_canonical": "Find all pairs $(m,n)$ of integers, $m ,n \\ge 2$ such that $mn - 1$ divides $n^3 - 1$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $ABC$ be a triangle inscribed in circle $(O),$ with its altitudes $BE, CF$ intersect at orthocenter $H$ ($E \\in AC, F \\in AB$). Let $M$ be the midpoint of $BC, K$ be the orthogonal projection of $H$ on $AM$. $EF$ intersects $BC$ at $P$. Let $Q$ be the intersection of tangent of $(O)$ which passes through $A$ with $BC, T$ be the reflection of $Q$ through $P$. Prove that $\\angle OKT = 90^o$.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $ABC$ be a triangle inscribed in the circle $(O)$. The bisector of $\\angle BAC$ cuts the circle $(O)$ again at $D$. Let $DE$ be the diameter of $(O)$. Let $G$ be a point on arc $AB$ which does not contain $C$. The lines $GD$ and $BC$ intersect at $F$. Let $H$ be a point on the line $AG$ such that $FH \\parallel AE$. Prove that the circumcircle of triangle $HAB$ passes through the orthocenter of triangle $HAC$.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $ABC$ be an acute, non isosceles triangle with $M, N, P$ are midpoints of $BC, CA, AB$, respectively. Denote $d_1$ as the line passes through $M$ and perpendicular to the angle bisector of $\\angle BAC$, similarly define for $d_2, d_3$. Suppose that $d_2 \\cap d_3 = D$, $d_3 \\cap d_1 = E,$ $d_1 \\cap d_2 = F$. Let $I, H$ be the incenter and orthocenter of triangle $ABC$. Prove that the circumcenter of triangle $DEF$ is the midpoint of segment $IH$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[\/i]. Two boxes [i]intersect[\/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \\ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\\not\\equiv j\\pm 1\\pmod n$.\r\n\r\n[i]Proposed by Gerhard Woeginger, Netherlands[\/i]",
"tags": [
"combinatorics",
"geometry"
]
},
{
"post_canonical": "Find all functions $f$ from the reals to the reals such that\r\n\r\n\\[f\\left(f(x)+y\\right)=2x+f\\left(f(y)-x\\right)\\]\r\n\r\nfor all real $x,y$.",
"tags": [
"algebra",
"function"
]
},
{
"post_canonical": "Prove the inequality:\n\\[\\sum_{i < j}{\\frac {a_{i}a_{j}}{a_{i} \\plus{} a_{j}}}\\leq \\frac {n}{2(a_{1} \\plus{} a_{2} \\plus{}\\cdots \\plus{} a_{n})}\\cdot \\sum_{i < j}{a_{i}a_{j}}\\]\nfor positive reals $ a_{1},a_{2},\\ldots,a_{n}$.\n\n[i]Proposed by Dusan Dukic, Serbia[\/i]",
"tags": [
"algebra",
"function",
"inequality"
]
},
{
"post_canonical": "Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\\left|a_{n+1}-a_{n+2}\\right|$ for all $n\\geq 0$, where $a_0$ and $a_1$ are two different positive reals.\n\nCan this sequence $a_0$, $a_1$, $a_2$, ... be bounded?\n\n[i]Proposed by Mihai B\u0103lun\u0103, Romania[\/i]",
"tags": [
"algebra"
]
},
{
"post_canonical": "Let $\\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \\tau(an)=n $ does not have a positive integer solution $n$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Find all monotonically increasing or monotonically decreasing functions $f: \\mathbb{R}_+\\to\\mathbb{R}_+$ which satisfy the equation $f\\left(xy\\right)\\cdot f\\left(\\frac{f\\left(y\\right)}{x}\\right)=1$ for any two numbers $x$ and $y$ from $\\mathbb{R}_+$.\r\n\r\nHereby, $\\mathbb{R}_+$ is the set of all positive real numbers.\r\n\r\n[i]Note.[\/i] A function $f: \\mathbb{R}_+\\to\\mathbb{R}_+$ is called [i]monotonically increasing[\/i] if for any two positive numbers $x$ and $y$ such that $x\\geq y$, we have $f\\left(x\\right)\\geq f\\left(y\\right)$.\r\nA function $f: \\mathbb{R}_+\\to\\mathbb{R}_+$ is called [i]monotonically decreasing[\/i] if for any two positive numbers $x$ and $y$ such that $x\\geq y$, we have $f\\left(x\\right)\\leq f\\left(y\\right)$.",
"tags": [
"algebra",
"function",
"inequality"
]
},
{
"post_canonical": "In the following, a [i]word[\/i] will mean a finite sequence of letters \"$a$\" and \"$b$\". The [i]length[\/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word.\r\n\r\nA word $w$ of length $\\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\\ell}$ in this order is called a [i]palindrome[\/i] if and only if $x_j=x_{\\ell+1-j}$ holds for every $j$ such that $1\\leq j\\leq\\ell$. For instance, $baaab$ is a palindrome; so is the empty word.\r\n\r\nFor two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$.\r\n\r\nLet $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "The two circles $\\Gamma_1$ and $\\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\\Gamma_1$ in $C \\neq A$ and $\\Gamma_2$ in $D \\neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$.\nProve that the four points $O_1,O_2,B$ and $X$ are concyclic. ",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\\neq j$.\n\nProve the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \\[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\\] are either all negative, all positive, or all zero.\n\n[i]Proposed by Kiran Kedlaya, USA[\/i]",
"tags": [
"algebra",
"geometry"
]
},
{
"post_canonical": "Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\\left(a\\right)$:\n\n(i) move the last digit of $a$ to the first position to obtain the numb er $b$;\n(ii) square $b$ to obtain the number $c$;\n(iii) move the first digit of $c$ to the end to obtain the number $d$.\n\n(All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.)\n\nFind all numbers $a$ for which $d\\left( a\\right) =a^2$.\n\n[i]Proposed by Zoran Sunic, USA[\/i]",
"tags": [
"combinatorics",
"number theory"
]
},
{
"post_canonical": "Consider the real number axis (i. e. the $x$-axis of a Cartesian coordinate system). We mark the points $1$, $2$, ..., $2n$ on this axis. A flea starts at the point $1$. Now it jumps along the real number axis; it can jump only from a marked point to another marked point, and it doesn't visit any point twice. After the ($2n-1$)-th jump, it arrives at a point from where it cannot jump any more after this rule, since all other points are already visited. Hence, with its $2n$-th jump, the flea breaks this rule and gets back to the point $1$. Assume that the sum of the (non-directed) lengths of the first $2n-1$ jumps of the flea was $n\\left(2n-1\\right)$. Show that the length of the last ($2n$-th) jump is $n$.",
"tags": [
"geometry",
"number theory"
]
},
{
"post_canonical": "Consider a polynomial $P(x) = \\prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \\ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \\geq N$ the number $P(x)$ is divisible by a prime number greater than 20.\n\n[i]Proposed by Luxembourg[\/i]",
"tags": [
"algebra",
"combinatorics",
"number theory",
"polynomial"
]
},
{
"post_canonical": "The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader\u2019s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader\u2019s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?\n",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Let $ ABC$ be an acute triangle, and $ M_a$, $ M_b$, $ M_c$ be the midpoints of the sides $ a$, $ b$, $ c$. The perpendicular bisectors of $ a$, $ b$, $ c$ (passing through $ M_a$, $ M_b$, $ M_c$) intersect the boundary of the triangle again in points $ T_a$, $ T_b$, $ T_c$. Show that if the set of points $ \\left\\{A,B,C\\right\\}$ can be mapped to the set $ \\left\\{T_a, T_b, T_c\\right\\}$ via a similitude transformation, then two feet of the altitudes of triangle $ ABC$ divide the respective triangle sides in the same ratio. (Here, \"ratio\" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?",
"tags": [
"geometry"
]
},
{
"post_canonical": "Find the largest possible integer $k$, such that the following statement is true: \nLet $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain\n\\[ \\left. \\begin{array}{rcl}\n & b_1 \\leq b_2\\leq\\ldots\\leq b_{2009} & \\textrm{the lengths of the blue sides }\\\\\n & r_1 \\leq r_2\\leq\\ldots\\leq r_{2009} & \\textrm{the lengths of the red sides }\\\\\n \\textrm{and } & w_1 \\leq w_2\\leq\\ldots\\leq w_{2009} & \\textrm{the lengths of the white sides }\\\\\n \\end{array}\\right.\\]\nThen there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$.\n\n[i]Proposed by Michal Rolinek, Czech Republic[\/i]",
"tags": [
"algebra",
"inequality"
]
},
{
"post_canonical": "Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram.\nProve that $GR=GS$.\n\n[i]Proposed by Hossein Karke Abadi, Iran[\/i]",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $a \\in \\mathbb{R}.$ Show that for $n \\geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \\frac{1}{\\sqrt[n]{n}}.$",
"tags": [
"algebra",
"polynomial"
]
},
{
"post_canonical": "A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off.\n\n[i]Proposed by Australia[\/i]",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.\r\nSuppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.",
"tags": [
"algebra",
"number theory",
"polynomial"
]
},
{
"post_canonical": "For any positive integer $n$, let $w\\left(n\\right)$ denote the number of different prime divisors of the number $n$. (For instance, $w\\left(12\\right)=2$.) Show that there exist infinitely many positive integers $n$ such that $w\\left(n\\right)<w\\left(n+1\\right)<w\\left(n+2\\right)$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Find all real solutions $x$ of the equation\r\n\r\n$\\cos\\cos\\cos\\cos x=\\sin\\sin\\sin\\sin x$.\r\n\r\n(Angles are measured in radians.)",
"tags": [
"algebra",
"function",
"trigonometry"
]
},
{
"post_canonical": "A sequence $x_1, x_2, \\ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \\geq 1.$ Prove that $\\forall n \\geq 1$ $x_1 + x_2 + \\ldots + x_n \\geq 0.$\n\n[i]Proposed by Gerhard W\u00f6ginger, Austria[\/i]",
"tags": [
"algebra",
"inequality"
]
},
{
"post_canonical": "Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \\in \\{1, 2, 3, \\ldots, 1000\\}$ and which are divisible by 121.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $P$ denote the set of all ordered pairs $ \\left(p,q\\right)$ of nonnegative integers. Find all functions $f: P \\rightarrow \\mathbb{R}$ satisfying\n\\[ f(p,q) \\equal{} \\begin{cases} 0 & \\text{if} \\; pq \\equal{} 0, \\\\\n1 \\plus{} \\frac{1}{2} f(p+1,q-1) \\plus{} \\frac{1}{2} f(p-1,q+1) & \\text{otherwise} \\end{cases}\n\\]\n\nCompare IMO shortlist problem 2001, algebra A1 for the three-variable case.",
"tags": [
"algebra",
"function"
]
},
{
"post_canonical": "Let $a_1,a_2,\\ldots a_n,k$, and $M$ be positive integers such that\n$$\\frac{1}{a_1}+\\frac{1}{a_2}+\\cdots+\\frac{1}{a_n}=k\\quad\\text{and}\\quad a_1a_2\\cdots a_n=M.$$\nIf $M>1$, prove that the polynomial\n$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\\cdots (x+a_n)$$\nhas no positive roots.",
"tags": [
"algebra",
"geometry",
"polynomial"
]
},
{
"post_canonical": "$ABCD$ is a cyclic quadrilateral and $\\omega$ its circumcircle. The perpendicular line to $AC$ at $D$ intersects $AC$ at $E$ and $\\omega$ at F. Denote by $\\ell$ the perpendicular line to $BC$ at $F$. The perpendicular line to $\\ell$ at A intersects $\\ell$ at $G$ and $\\omega$ at $H$. Line $GE$ intersects $FH$ at $I$ and $CD$ at $J$. Prove that points $C, F, I$ and $J$ are concyclic",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $ABC$ be a triangle with $M, N, P$ as midpoints of the segments $BC, CA,AB$ respectively. Suppose that $I$ is the intersection of angle bisectors of $\\angle BPM, \\angle MNP$ and $J$ is the intersection of angle bisectors of $\\angle CN M, \\angle MPN$. Denote $(\\omega_1)$ as the circle of center $I$ and tangent to $MP$ at $D$, $(\\omega_2)$ as the circle of center $J$ and tangent to $MN$ at $E$.\na) Prove that $DE$ is parallel to $BC$.\nb) Prove that the radical axis of two circles $(\\omega_1), (\\omega_2)$ bisects the segment $DE$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Consider a regular $2^k$-gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$-gon.\n\n[i]Proposed by Hesam Rajabzade[\/i]",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $ ABC$ be a triangle and $ A'$ , $ B'$ and $ C'$ lie on $ BC$ , $ CA$ and $ AB$ respectively such that the incenter of $ A'B'C'$ and $ ABC$ are coincide and the inradius of $ A'B'C'$ is half of inradius of $ ABC$ . Prove that $ ABC$ is equilateral .",
"tags": [
"geometry",
"trigonometry"
]
},
{
"post_canonical": "Let $f:\\mathbb N\\rightarrow\\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\\dots,p_n$ and natural numbers $s_1,s_2,\\dots,s_n$ such that for each $1\\leq i\\leq n$ the set $\\{f(p_ir+s_i)|r=1,2,\\dots\\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that\n\\[f(a+1), f(a+2), \\dots, f(a+n)\\]\nform an arithmetic progression.",
"tags": [
"function",
"number theory"
]
},
{
"post_canonical": "The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC,AC,AB$ at $A_{1},B_{1},C_{1}$ .\nlet $AI,BI,CI$ meets $BC,AC,AB$ at $A_{2},B_{2},C_{2}$.\nlet $A'$ is a point on $AI$ such that $A_{1}A'\\perp B_{2}C_{2}$ .$B',C'$ respectively.\nprove that two triangle $A'B'C',A_{1}B_{1}C_{1}$ are equal.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "$ABCD$ is a trapezoid with $AB \\parallel CD$. The diagonals intersect at $P$. Let $\\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\\omega _3$ is the circumcircle of triangle $BPC$.\nProve that the common chord of circles $\\omega _1,\\omega _3$ and the common chord of circles $\\omega _2, \\omega _3$ intersect each other on $AD$.\n\n[i]Proposed by Kasra Ahmadi[\/i]",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "$a,b,c,d$ are positive numbers such that $\\sum_{cyc} \\frac{1}{ab} =1$. Prove that :\n$abcd+16 \\geq 8 \\sqrt{(a+c)(\\frac{1}{a} + \\frac{1}{c})}+8\\sqrt{(b+d)(\\frac{1}{b}+\\frac{1}{d})}$",
"tags": [
"inequality"
]
},
{
"post_canonical": "Point $A$ is outside of a given circle $\\omega$. Let the tangents from $A$ to $\\omega$ meet $\\omega$ at $S, T$ points $X, Y$ are midpoints of $AT, AS$ let the tangent from $X$ to $\\omega$ meet $\\omega$ at $R\\neq T$. points $P, Q$ are midpoints of $XT, XR$ let $XY\\cap PQ=K, SX\\cap TK=L$ prove that quadrilateral $KRLQ$ is cyclic. ",
"tags": [
"geometry"
]
},
{
"post_canonical": "Suppose that $p$ is a prime number.\r\nFind all natural numbers $n$ such that $p|\\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have\r\n\\[ n|a^{\\frac{\\varphi(n)}{p}}-1 \\]",
"tags": [
"function",
"number theory"
]
},
{
"post_canonical": "We have $ 100$ equal cubes. Player $ A$ has to paint the faces of the cubes, each white or black, such that every cube has at least one face of each colour, at least $ 50$ cubes have more than one black face and at least $ 50$ cubes have more than one white face .\r\n\r\nPlayer $ B$ has to place the coloured cubes in a table in a way that their bases form the frame that surrounds a $ 40*12$ rectangle. There are some faces that can not been seen because they are overlapped with other faces or based on the table, we call them invisible faces. On the other hand, the ones which can be seen are called visible faces. Prove that player $ B$ can always place the cubes in such a way that the number of visible faces is the the same as the number of invisible faces, despite the initial colouring of player $ A$\r\n\r\nNote: It is easy to see that in the configuration, each cube has three visible faces and three invisible faces",
"tags": [
"combinatorics",
"geometry"
]
},
{
"post_canonical": "Let $S$ be a string of $99$ characters, $66$ of which are $A$ and $33$ are $B$. We call $S$ [i]good[\/i] if, for each $n$ such that $1\\le n \\le 99$, the sub-string made from the first $n$ characters of $S$ has an odd number of distinct permutations. How many good strings are there? Which strings are good?",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$",
"tags": [
"modular arithmetic",
"number theory"
]
},
{
"post_canonical": "1. The transformation $ n \\to 2n \\minus{} 1$ or $ n \\to 3n \\minus{} 1$, where $ n$ is a positive integer, is called the 'change' of $ n$. Numbers $ a$ and $ b$ are called 'similar', if there exists such positive integer, that can be got by finite number of 'changes' from both $ a$ and $ b$. Find all positive integers 'similar' to $ 2005$ and less than $ 2005$.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.",
"tags": [
"algebra",
"function",
"geometry"
]
},
{
"post_canonical": "Prove that for any n natural, the number \\[ \\sum \\limits_{k=0}^{n} \\binom{2n+1}{2k+1} 2^{3k} \\]\r\ncannot be divided by $5$.",
"tags": [
"modular arithmetic",
"number theory"
]
},
{
"post_canonical": "Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.",
"tags": [
"number theory"
]
},
{
"post_canonical": "The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $ BC, CA, AB$ at $ A_1 , B_1 , C_1 $ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2 $. The line $B_1 C_1 $ meets the line $BC$ at $A_3 $ and the line $A_2 A_3 $ meets the circumcircle of $ABC$ at $A_4 (\\ne A_2 ) $. Define $B_4 , C_4 $ similarly. Prove that the lines $ AA_4 , BB_4 , CC_4 $ are concurrent.",
"tags": [
"circle",
"geometry",
"trigonometry"
]
},
{
"post_canonical": "Does there exist a polynomial $P(x)$ with integer coefficients such that $P(1+\\sqrt[3]{2})=1+\\sqrt[3]{2}$ and $P(1+\\sqrt5)=2+3\\sqrt5$?",
"tags": [
"algebra",
"polynomial"
]
},
{
"post_canonical": "Let $a$ be a real number. Suppose the function $f(x) = \\frac{a}{x-1} + \\frac{1}{x-2} + \\frac{1}{x-6}$ defined in the interval $3 < x < 5$ attains its maximum at $x=4$. Find the value of $a.$",
"tags": [
"algebra",
"function"
]
},
{
"post_canonical": "Determine all sequences $p_1, p_2, \\dots $ of prime numbers for which there exists an integer $k$ such that the recurrence relation \n\\[ p_{n+2} = p_{n+1} + p_n + k \\]\nholds for all positive integers $n$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Suppose, $x, y, z \\in \\mathbb{R}_+$ such that $xy+yz+zx=1$. Prove that, \\[x(1-y^2)(1-z^2)+y(1-z^2)(1-x^2)+z(1-x^2)(1-y^2)\\leq \\frac{4\\sqrt{3}}{9}\\]",
"tags": [
"inequality"
]
},
{
"post_canonical": "Given that $\\{a_n\\}$ is a sequence of integers satisfying the following condition for all positive integral values of $n$: $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2016$. Find all possible values of $a_1$ and $a_2$",
"tags": [
"algebra"
]
},
{
"post_canonical": "Let $k>1$ be an integer. A function $f:\\mathbb{N^*}\\to\\mathbb{N^*}$ is called $k$-[i]tastrophic[\/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$:\n\\[f_k(n)=\\underbrace{f\\circ f\\circ\\cdots \\circ f}_{k\\text{ times}}(n)\\]\nFor which $k$ does there exist a $k$-tastrophic function?",
"tags": [
"algebra",
"function"
]
},
{
"post_canonical": "In an acute-angled triangle $ABC$, $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ respectively, and $M$ is the midpoint of $AB$.\na) Prove that $MA_1$ is tangent to the circumcircle of triangle $A_1B_1C$.\nb) Prove that the circumcircles of triangles $A_1B_1C,BMA_1$, and $AMB_1$ have a common point.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "If $n$ is a positive integer, let $A = \\{n,n+1,...,n+17 \\}$.\r\nDoes there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?",
"tags": [
"modular arithmetic",
"number theory"
]
},
{
"post_canonical": "$E$ and $F$ are interior points of convex quadrilateral $ABCD$ such that $AE = BE$, $CE = DE$, $\\angle AEB = \\angle CED$, $AF = DF$, $BF = CF$, $\\angle AFD = \\angle BFC$. Prove that $\\angle AFD + \\angle AEB = \\pi$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "The quadrilateral $ABCD$ is inscribed in circle $\\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "In acute triangle $ ABC$, show that:\r\n\r\n$ \\sin^3{A}\\cos^2{(B \\minus{} C)} \\plus{} \\sin^3{B}\\cos^2{(C \\minus{} A)} \\plus{} \\sin^3{C}\\cos^2{(A \\minus{} B)} \\leq 3\\sin{A} \\sin{B} \\sin{C}$\r\n\r\nand find out when the equality holds.",
"tags": [
"geometry",
"inequality",
"trigonometry"
]
},
{
"post_canonical": "Circle $ O$ is inscribed in a trapzoid $ ABCD$, $ \\angle{A}$ and $ \\angle{B}$ are all acute angles. A line through $ O$ intersects $ AD$ at $ E$ and $ BC$ at $ F$, and satisfies the following conditions:\r\n\r\n(1) $ \\angle{DEF}$ and $ \\angle{CFE}$ are acute angles.\r\n\r\n(2) $ AE\\plus{}BF\\equal{}DE\\plus{}CF$.\r\n\r\nLet $ AB\\equal{}a$, $ BC\\equal{}b$, $ CD\\equal{}c$, then use $ a,b,c$ to express $ AE$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Find all the pairs of positive integers $ (a,b)$ such that $ a^2 \\plus{} b \\minus{} 1$ is a power of prime number $ ; a^2 \\plus{} b \\plus{} 1$ can divide $ b^2 \\minus{} a^3 \\minus{} 1,$ but it can't divide $ (a \\plus{} b \\minus{} 1)^2.$",
"tags": [
"modular arithmetic",
"number theory"
]
},
{
"post_canonical": "Given a real number $\\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \\lambda BA$, $CV = \\lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \\lambda UV$. Find the locus of point $Q$.",
"tags": [
"circle",
"geometry",
"trigonometry"
]
},
{
"post_canonical": "In $\\triangle ABC$ we have $BC>CA>AB$. The nine point circle is tangent to the incircle, $A$-excircle, $B$-excircle and $C$-excircle at the points $T,T_A,T_B,T_C$ respectively. Prove that the segments $TT_B$ and lines $T_AT_C$ intersect each other.",
"tags": [
"geometry",
"trigonometry"
]
},
{
"post_canonical": "Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\\Gamma$ such that $PQ \\parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\\triangle KJM$ is an isosceles triangle.",
"tags": [
"circle",
"geometry",
"trigonometry"
]
},
{
"post_canonical": "Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Given a rectangle $ ABCD,$ let $ AB \\equal{} b, AD \\equal{} a ( a\\geq b),$ three points $ X,Y,Z$ are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by $ a,b$)",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $k$ be a positive integer. Prove that one can partition the set $\\{ 0,1,2,3, \\cdots ,2^{k+1}-1 \\}$ into two disdinct subsets $\\{ x_1,x_2, \\cdots, x_{2k} \\}$ and $\\{ y_1, y_2, \\cdots, y_{2k} \\}$ such that $\\sum_{i=1}^{2^k} x_i^m =\\sum_{i=1}^{2^k} y_i^m$ for all $m \\in \\{ 1,2, \\cdots, k \\}$.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \\neq n.$ Prove that if $ b^m \\minus{} 1$ and $ b^n \\minus{} 1$ have the same prime divisors, then $ b \\plus{} 1$ is a power of 2.",
"tags": [
"number theory"
]
},
{
"post_canonical": "A triangle of sides $\\frac{3}{2}, \\frac{\\sqrt{5}}{2}, \\sqrt{2}$ is folded along a variable line perpendicular to the side of $\\frac{3}{2}.$ Find the maximum value of the coincident area.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Find $k \\in \\mathbb{N}$ such that\n\n[b]a.)[\/b] For any $n \\in \\mathbb{N}$, there does not exist $j \\in \\mathbb{Z}$ which satisfies the conditions $0 \\leq j \\leq n - k + 1$ and $\\left(\n\\begin{array}{c}\nn\\\\\nj\\end{array} \\right), \\left( \\begin{array}{c}\nn\\\\\nj + 1\\end{array} \\right), \\ldots, \\left( \\begin{array}{c}\nn\\\\\nj + k - 1\\end{array} \\right)$ forms an arithmetic progression.\n\n[b]b.)[\/b] There exists $n \\in \\mathbb{N}$ such that there exists $j$ which satisfies $0 \\leq j \\leq n - k + 2$, and $\\left(\n\\begin{array}{c}\nn\\\\\nj\\end{array} \\right), \\left( \\begin{array}{c}\nn\\\\\nj + 1\\end{array} \\right), \\ldots , \\left( \\begin{array}{c}\nn\\\\\nj + k - 2\\end{array} \\right)$ forms an arithmetic progression.\n\nFind all $n$ which satisfies part [b]b.)[\/b]",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Let $ ABC$ be a triangle. Point $ D$ lies on its sideline $ BC$ such that $ \\angle CAD \\equal{} \\angle CBA.$ Circle $ (O)$ passing through $ B,D$ intersects $ AB,AD$ at $ E,F$, respectively. $ BF$ meets $ DE$ at $ G$.Denote by$ M$ the midpoint of $ AG.$ Show that $ CM\\perp AO.$",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $ \\alpha,\\beta$ be real numbers satisfying $ 1 < \\alpha < \\beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \\alpha\\le \\frac {x}{y}\\le\\beta.$",
"tags": [
"combinatorics",
"function"
]
},
{
"post_canonical": "Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\\minus{}1,$ then $ a^b > \\frac {3^n}{n}.$",
"tags": [
"inequality",
"number theory"
]
},
{
"post_canonical": "$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of:\r\n\\[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \\]",
"tags": [
"inequality"
]
},
{
"post_canonical": "In triangle $ABC$, $AB > BC > CA$, $AB=6$, $\\angle{B}-\\angle{C}=90^o$. The incircle touches $BC$ at $E$ and $EF$ is a diameter of the incircle. Radical $AF$ intersect $BC$ at $D$. $DE$ equals to the circumradius of $\\triangle{ABC}$. Find $BC$ and $AC$.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "There are $n$($n\\geq 3$) circles in the plane, all with radius $1$. In among any three circles, at least two have common point(s), then the total area covered by these $n$ circles is less than $35$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $F$ be the set of all polynomials $\\Gamma$ such that all the coefficients of $\\Gamma (x)$ are integers and $\\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\\Gamma \\in F$ for which $\\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.",
"tags": [
"algebra",
"function",
"polynomial"
]
},
{
"post_canonical": "Let side $BC$ of $\\bigtriangleup ABC$ be the diameter of a semicircle which cuts $AB$ and $AC$ at $D$ and $E$ respectively. $F$ and $G$ are the feet of the perpendiculars from $D$ and $E$ to $BC$ respectively. $DG$ and $EF$ intersect at $M$. Prove that $AM \\perp BC$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "The circle $\\Gamma$ through $A$ of triangle $ABC$ meets sides $AB,AC$ at $E$,$F$ respectively, and circumcircle of $ABC$ at $P$. Prove: Reflection of $P$ across $EF$ is on $BC$ if and only if $\\Gamma$ passes through $O$ (the circumcentre of $ABC$).",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $x_1,x_2,\\cdots,x_n$ $(n\\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\\frac{x_2}{2},\\cdots,\\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that\n\\[ \\frac{\\sum_{i=1}^{n} x_i }{n\\left (\\prod_{i=1}^{n}x_i \\right )^{\\frac{1}{n}}}\\le \\frac{n+1}{2\\sqrt[n]{n!}}\\]",
"tags": [
"inequality"
]
},
{
"post_canonical": "The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \\cap BD$. The circumecentres of $\\triangle{OAB}$, $\\triangle{OBC}$, $\\triangle{OCD}$ and $\\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively. \r\n\r\nProve that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $\\angle XOY = \\frac{\\pi}{2}$; $P$ is a point inside $\\angle XOY$ and we have $OP = 1; \\angle XOP = \\frac{\\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $\\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\\parallel IM$ and $CE\\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\\triangle ABC$.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\\angle ADC$ is acute and $\\frac{AB}{BC}=\\frac{DA}{CD}$. Let $\\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\\Gamma$ and inside $ABCD$.\nProve that $AE\\perp EC$ if and only if $\\frac{AE}{AB}-\\frac{ED}{AD}=1$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\\angle MAB=\\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\\triangle ABC,\\triangle ABP,\\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.",
"tags": [
"circle",
"geometry",
"trigonometry"
]
},
{
"post_canonical": "For non-negative real numbers $x_1, x_2, \\ldots, x_n$ which satisfy $x_1 + x_2 + \\cdots + x_n = 1$, find the largest possible value of $\\sum_{j = 1}^{n} (x_j^{4} - x_j^{5})$.",
"tags": [
"inequality"
]
},
{
"post_canonical": "Given a triangle $ ABC$ with angle $ C \\geq 60^{\\circ}$. Prove that:\r\n \r\n$ \\left(a \\plus{} b\\right) \\cdot \\left(\\frac {1}{a} \\plus{} \\frac {1}{b} \\plus{} \\frac {1}{c} \\right) \\geq 4 \\plus{} \\frac {1}{\\sin\\left(\\frac {C}{2}\\right)}.$",
"tags": [
"geometry",
"inequality",
"trigonometry"
]
},
{
"post_canonical": "In a simple graph $G$, we call $t$ pairwise adjacent vertices a $t$[i]-clique[\/i]. If a vertex is connected with all other vertices in the graph, we call it a [i]central[\/i] vertex. Given are two integers $n,k$ such that $\\dfrac {3}{2} \\leq \\dfrac{1}{2} n < k < n$. Let $G$ be a graph on $n$ vertices such that\n[b](1)[\/b] $G$ does not contain a $(k+1)$-[i]clique[\/i];\n[b](2)[\/b] if we add an arbitrary edge to $G$, that creates a $(k+1)$-[i]clique[\/i].\nFind the least possible number of [i]central[\/i] vertices in $G$.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \\geq N$, one has $f(n) \\leq n^{0.01}$. ",
"tags": [
"modular arithmetic",
"number theory"
]
},
{
"post_canonical": "1. A finite sequence of integers $a_0,a_1,...,a_n$ is called quadratic if $|a_k -a_{k-1}| = k^2$\nfor $n\\geq k\\geq1$.\n(a) Prove that for any two integers $b$ and $c$, there exist a natural number $n$ and a quadratic sequence \n with $a_0 = b$ and $a_n =c$.\n(b) Find the smallest natural number $n$ for which there exists a quadratic sequence\nwith $a_0 = 0$ and $a_n = 1997$",
"tags": [
"algebra"
]
},
{
"post_canonical": "The points $P$ and $Q$ lie on the diagonals $AC$ and $BD$, respectively, of a quadrilateral $ABCD$ such that $\\frac{AP}{AC} + \\frac{BQ}{BD} =1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$, and $CNP$ are concurrent.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \\in \\{1,2, \\cdots, n \\}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Cut $2003$ disjoint rectangles from an acute-angled triangle $ABC$, such that any of them has a parallel side to $AB$ and the sum of their areas is maximal.",
"tags": [
"geometry"
]
},
{
"post_canonical": "On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "Solve system of equation $$8(x^3+y^3+z^3)=73$$ $$2(x^2+y^2+z^2)=3(xy+yz+zx)$$ $$xyz=1$$ in set $\\mathbb{R}^3$\n",
"tags": [
"algebra"
]
},
{
"post_canonical": "Denote by $M$ and $N$ feets of perpendiculars from $A$ to angle bisectors of exterior angles at $B$ and $C,$ in triangle $\\triangle ABC.$ Prove that the length of segment $MN$ is equal to semiperimeter of triangle $\\triangle ABC.$",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $Z$ shape be a shape such that it covers $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$ and $(i+2,j+2)$ where $(i,j)$ stands for cell in $i$-th row and $j$-th column on an arbitrary table. At least how many $Z$ shapes is necessary to cover one $8 \\times 8$ table if every cell of a $Z$ shape is either cell of a table or it is outside the table (two $Z$ shapes can overlap and $Z$ shapes can rotate)?",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Let $ABCD$ be a quadrilateral inscribed in circle $k$. Lines $AB$ and $CD$ intersect at point $E$ such that $AB=BE$. Let $F$ be the intersection point of tangents on circle $k$ in points $B$ and $D$, respectively. If the lines $AB$ and $DF$ are parallel, prove that $A$, $C$ and $F$ are collinear. \n\n\n",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "In triangle $ABC$ it holds $|BC|= \\frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.",
"tags": [
"circle",
"geometry"
]
},
{
"post_canonical": "In a convex quadrilateral $ ABCD, $ let $ A\u2019,B\u2019 $ be the orthogonal projections to $ CD $ of $ A, $ respectively, $ B. $\n\n[b]a)[\/b] Assuming that $ BB\u2019\\le AA\u2019 $ and that the perimeter of $ ABCD $ is $ (AB+CD)\\cdot BB\u2019, $ is $ ABCD $ necessarily a trapezoid?\n[b]b)[\/b] The same question with the addition that $ \\angle BAD $ is obtuse.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $ABC$ be a triangle and let $X$,$Y$,$Z$ be interior points on the sides $BC$, $CA$, $AB$, respectively. Show that the magnified image of the triangle $XYZ$ under a homothety of factor $4$ from its centroid covers at least one of the vertices $A$, $B$, $C$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Find all sets $A$ and $B$ that satisfy the following conditions:\r\n\r\na) $A \\cup B= \\mathbb{Z}$;\r\n\r\nb) if $x \\in A$ then $x-1 \\in B$;\r\n\r\nc) if $x,y \\in B$ then $x+y \\in A$.\r\n\r\n[i]Laurentiu Panaitopol[\/i]",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\\{ A,B,C,D,M,N\\}$. Find the least number $k$ such that $s(M,N)\\le k$, for all points $M,N$.\n\n[i]Dinu \u0218erb\u0103nescu[\/i]",
"tags": [
"combinatorics",
"geometry"
]
},
{
"post_canonical": "Show that if $a,b,c$ are complex numbers that such that\n\\[ (a+b)(a+c)=b \\qquad (b+c)(b+a)=c \\qquad (c+a)(c+b)=a\\]\nthen $a,b,c$ are real numbers.",
"tags": [
"algebra",
"trigonometry"
]
},
{
"post_canonical": "Let $n$ be a positive integer and let $x_1, x_2, \\ldots, x_n$ be positive real numbers such that $x_1x_2 \\cdots x_n = 1$. Prove that \\[\\displaystyle\\sum_{i=1}^n x_i^n (1 + x_i) \\geq \\dfrac{n}{2^{n-1}} \\prod_{i=1}^n (1 + x_i).\\]\n\n[i]IMO Shortlist[\/i]",
"tags": [
"inequality"
]
},
{
"post_canonical": "Let $P$ be a point in the plane and let $\\gamma$ be a circle which does not contain $P$. Two distinct variable lines $\\ell$ and $\\ell'$ through $P$ meet the circle $\\gamma$ at points $X$ and $Y$, and $X'$ and $Y'$, respectively. Let $M$ and $N$ be the antipodes of $P$ in the circles $PXX'$ and $PYY'$, respectively. Prove that the line $MN$ passes through a fixed point. \n\n[i]Mihai Chis[\/i]",
"tags": [
"geometry"
]
},
{
"post_canonical": "A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$. \n\n[i]AMM Magazine[\/i]",
"tags": [
"algebra",
"modular arithmetic",
"polynomial"
]
},
{
"post_canonical": "Let $\\{a_n\\}_{n\\geq 1}$ be a sequence with $a_1=1$, $a_2=4$ and for all $n>1$, \\[ a_{n} = \\sqrt{ a_{n-1}a_{n+1} + 1 } . \\] \r\na) Prove that all the terms of the sequence are positive integers.\r\n\r\nb) Prove that $2a_na_{n+1}+1$ is a perfect square for all positive integers $n$.\r\n\r\n[i]Valentin Vornicu[\/i]",
"tags": [
"algebra"
]
},
{
"post_canonical": "Fix a point $O$ in the plane and an integer $n\\geq 3$. Consider a \ffinite family $\\mathcal{D}$ of closed unit discs in the plane such that:\n(a) No disc in $\\mathcal{D}$ contains the point $O$; and\n(b) For each positive integer $k < n$, the closed disc of radius $k + 1$ centred at $O$ contains the centres of at least $k$ discs in $\\mathcal{D}$.\nShow that some line through $O$ stabs at least $\\frac{2}{\\pi} \\log \\frac{n+1}{2}$ discs in $\\mathcal{D}$.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "We are given in the plane a line $\\ell$ and three circles with centres $A,B,C$ such that they are all tangent to $\\ell$ and pairwise externally tangent to each other. Prove that the triangle $ABC$ has an obtuse angle and find all possible values of this this angle.\n\n[i]Mircea Becheanu[\/i]",
"tags": [
"geometry",
"inequality",
"trigonometry"
]
},
{
"post_canonical": "Let $VA_1A_2\\ldots A_n$ be a pyramid, where $n\\ge 4$. A plane $\\Pi$ intersects the edges $VA_1,VA_2,\\ldots, VA_n$ at the points $B_1,B_2,\\ldots,B_n$ respectively such that the polygons $A_1A_2\\ldots A_n$ and $B_1B_2\\ldots B_n$ are similar. Prove that the plane $\\Pi$ is parallel to the plane containing the base $A_1A_2\\ldots A_n$.\n\n[i]Laurentiu Panaitopol[\/i]",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $a>1$ be an odd positive integer. Find the least positive integer $n$ such that $2^{2000}$ is a divisor of $a^n-1$.\n\n[i]Mircea Becheanu [\/i]",
"tags": [
"modular arithmetic",
"number theory"
]
},
{
"post_canonical": "a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11;\r\n\r\nb) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11.",
"tags": [
"number theory"
]
},
{
"post_canonical": "On a $2004 \\times 2004$ chess table there are 2004 queens such that no two are attacking each other\\footnote[1]{two queens attack each other if they lie on the same row, column or direction parallel with on of the main diagonals of the table}.\r\n\r\nProve that there exist two queens such that in the rectangle in which the center of the squares on which the queens lie are two opposite corners, has a semiperimeter of 2004.",
"tags": [
"combinatorics",
"geometry",
"modular arithmetic"
]
},
{
"post_canonical": "Let $ABC$ be a triangle, and let $M$ be a point on the side $(AC)$ .The line through $M$ and parallel to $BC$ crosses $AB$ at $N$. Segments $BM$ and $CN$ cross at $P$, and the circles $BNP$ and $CMP$ cross again at $Q$. Show that angles $BAP$ and $CAQ$ are equal.",
"tags": [
"geometry"
]
},
{
"post_canonical": "A word of length $n$ is an ordered sequence $x_1x_2\\ldots x_n$ where $x_i$ is a letter from the set $\\{ a,b,c \\}$. Denote by $A_n$ the set of words of length $n$ which do not contain any block $x_ix_{i+1}, i=1,2,\\ldots ,n-1,$ of the form $aa$ or $bb$ and by $B_n$ the set of words of length $n$ in which none of the subsequences $x_ix_{i+1}x_{i+2}, i=1,2,\\ldots n-2,$ contains all the letters $a,b,c$.\nProve that $|B_{n+1}|=3|A_n|$.\n\n[i]Vasile Pop[\/i]",
"tags": [
"combinatorics",
"function"
]
},
{
"post_canonical": "If $a_{1}$, $a_{2}$, $\\ldots$, $a_{n}\\geq 0$ are such that \\[a_{1}^{2}+\\cdots+a_{n}^{2}=1,\\]\r\n then find the maximum value of the product $(1-a_{1})\\cdots (1-a_{n})$.",
"tags": [
"inequality"
]
},
{
"post_canonical": "Prove that the function $f : \\mathbb{N}\\longrightarrow \\mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.",
"tags": [
"function",
"inequality",
"number theory"
]
},
{
"post_canonical": "In a circle with center $O$ is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $ABCD$ be a circumscribed quadrilateral such that $AD>\\max\\{AB,BC,CD\\}$, $M$ be the common point of $AB$ and $CD$ and $N$ be the common point of $AC$ and $BD$. Show that \\[90^{\\circ}<m(\\angle AND)<90^{\\circ}+\\frac{1}{2}m(\\angle AMD).\\]\n\nFixed, thank you Luis.",
"tags": [
"geometry",
"inequality"
]
},
{
"post_canonical": "The quadrilateral $ ABCD$ inscribed in a circle wich has diameter $ BD$. Let $ A',B'$ are symmetric to $ A,B$ with respect to the line $ BD$ and $ AC$ respectively. If $ A'C \\cap BD \\equal{} P$ and $ AC\\cap B'D \\equal{} Q$ then prove that $ PQ \\perp AC$",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $n_1,\\ldots,n_k$ be positive integers, and define $d_1=1$ and $d_i=\\frac{(n_1,\\ldots,n_{i-1})}{(n_1,\\ldots,n_{i})}$, for $i\\in \\{2,\\ldots,k\\}$, where $(m_1,\\ldots,m_{\\ell})$ denotes the greatest common divisor of the integers $m_1,\\ldots,m_{\\ell}$. Prove that the sums \\[\\sum_{i=1}^k a_in_i\\] with $a_i\\in\\{1,\\ldots,d_i\\}$ for $i\\in\\{1,\\ldots,k\\}$ are mutually distinct $\\mod n_1$.",
"tags": [
"geometry",
"number theory"
]
},
{
"post_canonical": "Let $ABC$ be a triangle. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$; similarly, let $Q_1$ and $Q_2$ be points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ meet at $R$, and the circles $P_1P_2R$ and $Q_1Q_2R$ meet again at $S$, situated inside triangle $P_1Q_1R$. Finally, let $M$ be the midpoint of the side $AC$. Prove that the angles $P_1RS$ and $Q_1RM$ are equal.",
"tags": [
"geometry"
]
},
{
"post_canonical": "[color=darkblue]Let $ ABCD$ be a trapezoid with $ AB\\parallel CD$. Exterior equilateral triangles $ ABE$ and $ CDF$ are constructed. Prove that lines $ AC$, $ BD$ and $ EF$ are concurrent.[\/color]",
"tags": [
"geometry"
]
},
{
"post_canonical": "Determine a subset $ A\\subset \\mathbb{N}^*$ having $ 5$ different elements, so that the sum of the squares of its elements equals their product.\r\nDo not simply post the subset, show how you found it.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Positive numbers $\\alpha \u000b,\\beta \f, x_1, x_2,\\ldots, x_n$ ($n \\geq 1$) satisfy $x_1+x_2+\\cdots+x_n = 1$. Prove that\n\\[\\sum_{i=1}^{n} \\frac{x_i^3}{\\alpha x_i+\\beta x_{i+1}} \\geq \\frac{1}{n(\\alpha+\\beta)}.\\]\n[b]Note.[\/b] $x_{n+1}=x_1$.",
"tags": [
"inequality"
]
},
{
"post_canonical": "Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form \r\n$ P(X)\\equal{}X^{2n}\\plus{}(2n\\minus{}10)X^{2n\\minus{}1}\\plus{}a_2X^{2n\\minus{}2}\\plus{}...\\plus{}a_{2n\\minus{}2}X^2\\plus{}(2n\\minus{}10)X\\plus{}1$,\r\nif it is known that all the roots of them are positive reals.\r\n\r\n[i]Proposer[\/i]: [b]Baltag Valeriu[\/b]",
"tags": [
"algebra",
"polynomial"
]
},
{
"post_canonical": "Let $S$ be the set of all natural numbers with the property: the sum of the biggest three divisors of number $n$, different from $n$, is bigger than $n$. Determine the largest natural number $k$, which divides any number from $S$.\n(A natural number is a positive integer)",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.",
"tags": [
"circle",
"function",
"geometry",
"inequality",
"trigonometry"
]
},
{
"post_canonical": "Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Determine all integers $ n\\geq 2$ having the following property: for any integers $a_1,a_2,\\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \\leq i \\leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\\ldots,a_i+a_{i+1}+\\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.\n\n[i]Proposed by Warut Suksompong, Thailand[\/i]",
"tags": [
"number theory"
]
},
{
"post_canonical": "You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \\leq r \\leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "For each prime $p$, construct a graph $G_p$ on $\\{1,2,\\ldots p\\}$, where $m\\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "There are $2^{2018}$ positions on a circle numbered from $1$ to $2^{2018}$ in a clockwise manner. Initially, two white marbles are placed at positions $2018$ and $2019$. Before the game starts, Ping chooses to place either a black marble or a white marble at each remaining position. At the start of the game, Ping is given an integer $n$ ($0\\leq n\\leq 2018$) and two marbles, one black and one white. He will then move around the circle, starting at position $2n$ and moving clockwise by $2n$ positions at a time. At the starting position and each position he reaches, Ping must switch the marble at that position with a marble of the other color he carries. If he cannot do so at any position, he loses the game. Is there a way to place the $2^{2018}-2$ remaining marbles so that Ping will never lose the game regardless of the number $n$ and the number of rounds he moves around the circle?",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Let $n>1$ be a positive integer. Each cell of an $n\\times n$ table contains an integer. Suppose that the following conditions are satisfied:\n[list=1]\n[*] Each number in the table is congruent to $1$ modulo $n$.\n[*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$.\n[\/list]\nLet $R_i$ be the product of the numbers in the $i^{\\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\\text{th}}$ column. Prove that the sums $R_1+\\hdots R_n$ and $C_1+\\hdots C_n$ are congruent modulo $n^4$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \\geq 2$.",
"tags": [
"inequality"
]
},
{
"post_canonical": "A triangle $ABC$ with orthocenter $H$ is given. $P$ is a variable point on line $BC$. The perpendicular to $BC$ through $P$ meets $BH$, $CH$ at $X$, $Y$ respectively. The line through $H$ parallel to $BC$ meets $AP$ at $Q$. Lines $QX$ and $QY$ meet $BC$ at $U$, $V$ respectively. Find the shape of the locus of the incenters of the triangles $QUV$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $n>1$ be an integer. Find all $r\\in \\mathbb{R}$ so that the system of equations in real variables $x_1, x_2, \\dots, x_n$:\n\\begin{align*}\n&(r\\cdot x_1-x_2)(r\\cdot x_1-x_3)\\dots (r\\cdot x_1-x_n)=\\\\\n=&(r\\cdot x_2-x_1)(r\\cdot x_2-x_3)\\dots (r\\cdot x_2-x_n)=\\\\\n&\\qquad \\qquad \\qquad \\qquad \\vdots \\\\\n=&(r\\cdot x_n-x_1)(r\\cdot x_n-x_2)\\dots (r\\cdot x_n-x_{n-1})\n\\end{align*}\nhas a solution where the numbers $x_1, x_2, \\dots, x_n$ are pairwise distinct.",
"tags": [
"algebra"
]
},
{
"post_canonical": "Ayala and Barvaz play a game: Ayala initially gives Barvaz two $100\\times100$ tables of positive integers, such that the product of numbers in each table is the same. In one move, Barvaz may choose a row or column in one of the tables, and change the numbers in it (to some positive integers), as long as the total product remains the same. Barvaz wins if after $N$ such moves, he manages to make the two tables equal to each other, and otherwise Ayala wins. \na. For which values of $N$ does Barvaz have a winning strategy?\nb. For which values of $N$ does Barvaz have a winning strategy, if all numbers in Ayalah\u2019s tables must be powers of $2$?\n",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "A pair of positive integers $(a,b)$ is called an [b]average couple[\/b] if there exist positive integers $k$ and $c_1, \\dots, c_k$ for which \n\\[\\frac{c_1+c_2+\\cdots+c_k}{k}=a\\qquad \\text{and} \\qquad \\frac{s(c_1)+s(c_2)+\\cdots+s(c_k)}{k}=b\\]\nwhere $s(n)$ denotes the sum of digits of $n$ in decimal representation.\nFind the number of average couples $(a,b)$ for which $a,b<10^{10}$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "If $ p(x)$ is a polynomial, denote by $ p^n(x)$ the polynomial $ p(p(...(p(x))..)$, where $ p$ is iterated $ n$ times. Prove that the polynomial $ p^{2003}(x)\\minus{}2p^{2002}(x)\\plus{}p^{2001}(x)$ is divisible by $ p(x)\\minus{}x$",
"tags": [
"algebra",
"polynomial"
]
},
{
"post_canonical": "Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that\n\na) If $A\\subseteq S$, then $A\\in F$ if and only if $S\\setminus A\\notin F$;\n\nb) If $A\\subseteq B\\subseteq S$ and $B\\in F$, then $A\\in F$.\n\nDetermine if there must exist a function $f:S\\to\\mathbb{R}$ such that for every $A\\subseteq S$, $A\\in F$ if and only if\n\\[\\sum_{s\\in A}f(s)<\\sum_{s\\in S\\setminus A}f(s).\\]\n\n[i]Evan O'Dorney.[\/i]",
"tags": [
"combinatorics",
"function",
"modular arithmetic"
]
},
{
"post_canonical": "The integers $ 1,2,\\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\\minus{}b \\ge 2$ and replace them with $ a\\minus{}1$ and $ b\\plus{}1$[\/i]. Please, determine the maximum number of steps that can be done.\r\n[i]Yudi Satria, Jakarta[\/i]",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Consider the sequence of rational numbers defined by $x_1=\\frac{4}{3}$, and $x_{n+1}=\\frac{x_n^2}{x_n^2-x_n+1}$. Show that the nu,erator of the lowest term expression of each sum $x_1+x_2+...+x_k$ is a perfect square.",
"tags": [
"number theory"
]
},
{
"post_canonical": "In the vertexes of a regular $ 100$-gon we place the numbers from $ 1$ to $ 100$ , in some order, every number appearing exactly once. \r\nWe say that an arrangment of the numbers is happy if for every simmetry axis of the polygon, the numbers which are from one side of the axis are greater that their respective simmetrics (we don't take into consideration the numbers which are on the axis)\r\nFind all happy arrangments (If two happy arrangments are the same under the rotation they are considered as only one)",
"tags": [
"combinatorics",
"geometry"
]
},
{
"post_canonical": "Find the 3-digit number whose ratio with the sum of its digits it's minimal.",
"tags": [
"function",
"number theory"
]
},
{
"post_canonical": "Triangle $ABC$ is inscribed in circle $(O)$. $A$ varies on $(O)$ such that $AB>BC$. $M$ is the midpoint of $AC$. The circle with diameter $BM$ intersects $(O)$ at $R$. $RM$ intersects $(O)$ at $Q$ and intersects $BC$ at $P$. The circle with diameter $BP$ intersects $AB, BO$ at $K,S$ in this order.\na. Prove that $SR$ passes through the midpoint of $KP$.\nb. Let $N$ be the midpoint of $BC$. The radical axis of circles with diameters $AN, BM$ intersects $SR$ at $E$. Prove that $ME$ always passes through a fixed point.\n",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let be four positive integers $m, n, p, q$, with $p < m$ given and $q < n$. Take four points $A(0; 0), B(p; 0), C (m; q)$ and $D(m; n)$ in the coordinate plane. Consider the paths $f$ from $A$ to $D$ and the paths $g$ from $B$ to $C$ such that when going along $f$ or $g$, one goes only in the positive directions of coordinates and one can only change directions (from the positive direction of one axe coordinate into the the positive direction of the other axe coordinate) at the points with integral coordinates. Let $S$ be the number of couples $(f, g)$ such that $f$ and $g$ have no common points. Prove that\r\n\r\n\\[S = \\binom{n}{m+n} \\cdot \\binom{q}{m+q-p} - \\binom{q}{m+q} \\cdot \\binom{n}{m+n-p}.\\]",
"tags": [
"combinatorics",
"geometry"
]
},
{
"post_canonical": "On the sides of triangle $ABC$ take the points $M_1, N_1, P_1$ such that each line $MM_1, NN_1, PP_1$ divides the perimeter of $ABC$ in two equal parts ($M, N, P$ are respectively the midpoints of the sides $BC, CA, AB$).\r\n\r\n[b]I.[\/b] Prove that the lines $MM_1, NN_1, PP_1$ are concurrent at a point $K$.\r\n[b]II.[\/b] Prove that among the ratios $\\frac{KA}{BC}, \\frac{KB}{CA}, \\frac{KC}{AB}$ there exist at least a ratio which is not less than $\\frac{1}{\\sqrt{3}}$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that $| f(x) - P(x)| \\leq c \\cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.",
"tags": [
"algebra",
"function",
"polynomial"
]
},
{
"post_canonical": "Find all triangles $ABC$ for which $\\angle ACB$ is acute and the interior angle bisector of $BC$ intersects the trisectors $(AX, (AY$ of the angle $\\angle BAC$ in the points $N,P$ respectively, such that $AB=NP=2DM$, where $D$ is the foot of the altitude from $A$ on $BC$ and $M$ is the midpoint of the side $BC$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Is there $14$ consecutive positive integers such that each of these numbers is divisible by one of the prime numbers $p$ where $2\\leq p \\leq 11$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Show that any triangular prism of in\ufb01nite length can be cut by a plane such that the resulting intersection is an equilateral triangle.",
"tags": [
"geometry"
]
},
{
"post_canonical": "An $11\\times 11$ chess board is covered with one $\\boxed{ }$ shaped and forty $\\boxed{ }\\boxed{ }\\boxed{ }$ shaped tiles. Determine the squares where $\\boxed{}$ shaped tile can be placed.",
"tags": [
"combinatorics",
"geometry"
]
},
{
"post_canonical": "The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ with $S_{ABC} = S_{ADC}$ intersect at $E$. The lines through $E$ parallel to $AD$, $DC$, $CB$, $BA$\nmeet $AB$, $BC$, $CD$, $DA$ at $K$, $L$, $M$, $N$, respectively. Compute the ratio $\\frac{S_{KLMN}}{S_{ABC}}$",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $C',B',A'$ be points respectively on sides $AB,AC,BC$ of $\\triangle ABC$ satisfying $ \\tfrac{AB'}{B'C}= \\tfrac{BC'}{C'A}=\\tfrac{CA'}{A'B}=k$. Prove that the ratio of the area of the triangle formed by the lines $AA',BB',CC'$ over the area of $\\triangle ABC$ is $\\tfrac{(k-1)^2}{(k^2+k+1)}$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "A frog is jumping on $N$ stones which are numbered from $1$ to $N$ from left to right. The frog is jumping to the previous stone (to the left) with probability $p$ and is jumping to the next stone (to the right) with probability $1-p$. If the frog has jumped to the left from the leftmost stone or to the right from the rightmost stone, it will fall into the water. The frog is initially on the leftmost stone. If $p< \\tfrac 13$, show that the frog will fall into the water from the rightmost stone with a probability higher than $\\tfrac 12$.",
"tags": [
"algebra"
]
},
{
"post_canonical": "There are 2019 cards in a box. Each card has a number written on one of its sides and a letter on the other side. Amy and Ben play the following game: in the beginning Amy takes all the cards, places them on a line and then she flips as many cards as she wishes. Each time Ben touches a card he has to flip it and its neighboring cards. Ben is allowed to have as many as 2019 touches. Ben wins if all the cards are on the numbers' side, otherwise Amy wins. Determine who has a winning strategy.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "For all positive integers $a$ and $b$, we de\fne $a @ b = \\frac{a - b}{gcd(a, b)}$ .\nShow that for every integer $n > 1$, the following holds: \n$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers, \ndoes $a = b = c$ necessarily hold?\n(b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers, \ndoes $a = b = c$ necessarily hold?",
"tags": [
"algebra"
]
},
{
"post_canonical": "Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "In a quadrilateral $ABCD$ we have $\\angle A = \\angle C = 90^o$. Let $E$ be a point in the interior of $ABCD$. Let $M$ be the midpoint of $BE$. Prove that $\\angle ADB = \\angle EDC$ if and only if $|MA| = |MC|$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\\sqrt{x - y}$.",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\\angle ABC$ and $\\angle ADC$ are concurrent with $AC$.",
"tags": [
"geometry"
]
},
{
"post_canonical": "Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$",
"tags": [
"geometry"
]
},
{
"post_canonical": "For a positive integer $n$, a [i]sum-friendly odd partition[\/i] of $n$ is a sequence $(a_1, a_2, \\ldots, a_k)$ of odd positive integers with $a_1 \\le a_2 \\le \\cdots \\le a_k$ and $a_1 + a_2 + \\cdots + a_k = n$ such that for all positive integers $m \\le n$, $m$ can be [b]uniquely[\/b] written as a subsum $m = a_{i_1} + a_{i_2} + \\cdots + a_{i_r}$. (Two subsums $a_{i_1} + a_{i_2} + \\cdots + a_{i_r}$ and $a_{j_1} + a_{j_2} + \\cdots + a_{j_s}$ with $i_1 < i_2 < \\cdots < i_r$ and $j_1 < j_2 < \\cdots < j_s$ are considered the same if $r = s$ and $a_{i_l} = a_{j_l}$ for $1 \\le l \\le r$.) For example, $(1, 1, 3, 3)$ is a sum-friendly odd partition of $8$. Find the number of sum-friendly odd partitions of $9999$.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Determine all sequences $(x_1,x_2,\\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \\[\\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\\]\n\n[i]Proposed by Warut Suksompong, Thailand[\/i]",
"tags": [
"algebra"
]
},
{
"post_canonical": "Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \\neq c$ or $b \\neq d $, such that \\[ \\left| \\frac{a-b}{c-d} - 1 \\right|< \\frac{1}{100000}. \\]",
"tags": [
"algebra"
]
},
{
"post_canonical": "Let $n$ be a positive even integer, and let $c_1, c_2, \\dots, c_{n-1}$ be real numbers satisfying \\[ \\sum_{i=1}^{n-1} \\left\\lvert c_i-1 \\right\\rvert < 1. \\] Prove that \\[\n\t2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \\dots - c_1x^1 + 2\n\\] has no real roots.",
"tags": [
"algebra",
"polynomial"
]
},
{
"post_canonical": "A communications network consisting of some terminals is called a [i]$3$-connector[\/i] if among any three terminals, some two of them can directly communicate with each other. A communications network contains a [i]windmill[\/i] with $n$ blades if there exist $n$ pairs of terminals $\\{x_{1},y_{1}\\},\\{x_{2},y_{2}\\},\\ldots,\\{x_{n},y_{n}\\}$ such that each $x_{i}$ can directly communicate with the corresponding $y_{i}$ and there is a [i]hub[\/i] terminal that can directly communicate with each of the $2n$ terminals $x_{1}, y_{1},\\ldots,x_{n}, y_{n}$ . Determine the minimum value of $f (n)$, in terms of $n$, such that a $3$ -connector with $f (n)$ terminals always contains a windmill with $n$ blades.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:\n[list]\n[*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.\n[*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are\ntwo (not necessarily distinct) numbers from the first line.\n[*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.\n[\/list]\nDetermine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.",
"tags": [
"algebra"
]
},
{
"post_canonical": "Find all prime numbers $p,q$ such that: \n$$p^4+p^3+p^2+p=q^2+q$$",
"tags": [
"number theory"
]
},
{
"post_canonical": "All positive integers are coloured either red or green, such that the following conditions are satisfi\fed:\n- There are equally many red as green integers.\n- The sum of three (not necessarily distinct) red integers is red.\n- The sum of three (not necessarily distinct) green integers is green.\nFind all colourings that satisfy these conditions.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "[color=blue][size=150]PERU TST IMO - 2006[\/size]\nSaturday, may 20.[\/color]\r\n\r\n[b]Question 01[\/b]\r\n\r\nFind all $(x,y,z)$ positive integers, such that:\r\n\r\n$\\sqrt{\\frac{2006}{x+y}} + \\sqrt{\\frac{2006}{y+z}} + \\sqrt{\\frac{2006}{z+x}},$\r\n\r\nis an integer.\r\n\r\n---\r\n[url=http:\/\/www.mathlinks.ro\/Forum\/viewtopic.php?t=88509]Spanish version[\/url]\r\n\r\n$\\text{\\LaTeX}{}$ed by carlosbr",
"tags": [
"number theory"
]
},
{
"post_canonical": "Let $k$ be a positive number and $P$ a Polynomio with integer coeficients.\r\nProve that exists a $n$ positive integer such that:\r\n$P(1)+P(2)+\\dots+P(N)$ is divisible by $k$.",
"tags": [
"algebra",
"modular arithmetic"
]
},
{
"post_canonical": "Determine all integers $n \\geq 2$ for which the number $11111$ in base $n$ is a perfect square.",
"tags": [
"number theory"
]
},
{
"post_canonical": "On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Find all $n \\in N$ for which the value of the expression $x^n+y^n+z^n$ is constant for all $x,y,z \\in R$ such that $x+y+z=0$ and $xyz=1$.\n\nD. Bazylev",
"tags": [
"algebra"
]
},
{
"post_canonical": "In the sequence of digits $2,0,2,9,3,...$ any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers $2,0,1,5$ in that order occur in the sequence?\n\nFolklore",
"tags": [
"number theory"
]
},
{
"post_canonical": "Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \\times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\\mathcal{A}$.\n\nLet $\\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\\mathcal{B}$ than of arrangement $\\mathcal{A}$. Prove that arrangement $\\mathcal{B}$ can be obtained from $ \\mathcal{A}$ by performing a number of switches, defined as follows: \n\nA switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Call admissible a set $A$ of integers that has the following property:\nIf $x,y \\in A$ (possibly $x=y$) then $x^2+kxy+y^2 \\in A$ for every integer $k$.\nDetermine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.\n\n[i]Proposed by Warut Suksompong, Thailand[\/i]",
"tags": [
"number theory"
]
},
{
"post_canonical": "[b](i)[\/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 3-clique (3 nodes joined pairwise by edges).\r\n[b](ii)[\/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 4-clique (4 nodes joined pairwise by edges).",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[\/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[\/i] if no road belongs to two or more paths in the collection.\n\nLet $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.\n\n[i]Proposed by Warut Suksompong, Thailand[\/i]",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "A positive integer $n$ is called [i]naughty[\/i] if it can be written in the form $n=a^b+b$ with integers $a,b \\geq 2$.\nIs there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?",
"tags": [
"number theory"
]
},
{
"post_canonical": "For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "Define a $ k$-[i]clique[\/i] to be a set of $ k$ people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "There are $32$ students in the class with $10$ interesting group. Each group contains exactly $16$ students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their $interests-disparity$. Define $S$ as the sum of the $interests-disparity$ of all the couples, $\\binom{32}{2}\\left ( =\\: 496 \\right )$ ones in total. Determine the minimal possible value of $S$.",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$, $|E|>2018$, find the minimum of $|E|$ .",
"tags": [
"combinatorics"
]
},
{
"post_canonical": "We call a triangle $\\triangle ABC$, $Q$-angled if $\\tan\\angle A,\\tan\\angle B,\\tan\\angle C \\in \\mathbb{Q}$, where $\\angle A,\\angle B ,\\angle C$ are the interior angles of the triangle $\\triangle ABC$.\n$a)$ Prove that $Q$-angled triangles exist;\n$b)$ Let triangle $\\triangle ABC$ be $Q$-angled. Prove that for any non-negative integer $n$, numbers of the form\n$E_n=\\sin^n\\angle A \\sin^n\\angle B \\sin^n\\angle C + \\cos^n\\angle A\\cos^n\\angle B\\cos^n\\angle C$ are rational.",
"tags": [
"algebra",
"trigonometry"
]
},
{
"post_canonical": "Let $p_1, p_2, p_3, \\ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between 0 and 1. For positive integer $k$, define\r\n\\[ x_k = \\begin{cases} 0 & \\mbox{if} \\; x_{k-1} = 0, \\\\[.1in] {\\displaystyle \\left\\{ \\frac{p_k}{x_{k-1}} \\right\\}} & \\mbox{if} \\; x_{k-1} \\neq 0, \\end{cases} \\]\r\nwhere $\\{x\\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \\lfloor x \\rfloor$ where $\\lfloor x \\rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \\ldots$ eventually becomes 0.",
"tags": [
"function",
"modular arithmetic"
]
},
{
"post_canonical": "Let $x \\neq 0$ be a real number satisfying $ax^2+bx+c=0$ with $a,b,c \\in \\mathbb{Z}$ obeying $|a|+|b|+|c| > 1$. Then prove \\[ |x| \\geq \\frac{1}{|a|+|b|+|c|-1}. \\]",
"tags": [
"inequality",
"polynomial"
]
},
{
"post_canonical": "Let $P$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have positive imaginary part, and suppose that $P=r(\\cos \\theta^\\circ+i\\sin \\theta^\\circ),$ where $0<r$ and $0\\le \\theta <360.$ Find $\\theta.$",
"tags": [
"polynomial",
"trigonometry"
]
},
{
"post_canonical": "We are placing $n$ integers whose sum is $94$ over a circle such that each number is equal to the absolute value of the difference of (clockwise) next two numbers. What is the largest $n$ that makes such placing possible?\n\n$ \n\\textbf{(A)}\\ 188\n\\qquad\\textbf{(B)}\\ 186\n\\qquad\\textbf{(C)}\\ 141\n\\qquad\\textbf{(D)}\\ 100\n\\qquad\\textbf{(E)}\\ 47\n$",
"tags": []
},
{
"post_canonical": "$z$ is a complex number and $|z|=1$ and $z^2\\neq 1$. Then $\\frac{z}{1-z^2}$ lies on \n[list=1]\n[*] A line not passing through origin\n[*] $|z|=2$\n[*] $x$-axis\n[*] $y$-axis\n[\/list]",
"tags": []
},
{
"post_canonical": "There are 168 primes below 1000. Then sum of all primes below 1000 is \n[list=1]\n[*] 11555\n[*] 76127\n[*] 57298\n[*] 81722\n[\/list]",
"tags": []
},
{
"post_canonical": "Find the limit $\\lim \\limits_{n \\to \\infty} \\sin{n!}$\n[list=1]\n[*] 1\n[*] 0\n[*] $\\frac{\\pi}{4}$\n[*] None of the above\n[\/list]",
"tags": []
}
]
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