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a.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"다음 조건을 만족시키는 두 자연수 $a$, $b$의 모든 순서쌍 $(a, b)$의 개수를 구하여라.\n",
"\n",
"- $1 \\le a \\le 10$, $1 \\le b \\le 100$\n",
"- 곡선 $y=2^x$이 원 $(x-a)^2 + (y-b)^2 = 1$과 만나지 않는다\n",
"- 곡선 $y=2^x$이 원 $(x-a)^2 + (y-b)^2 = 4$와 적어도 한 점에서 만난다"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Solution\n",
"\n",
"먼저 임의의 점 $(p, q)$와 $y=2^x$와의 거리를 측정해보도록 하자. 이는 $y=2^x$라는 constraint가 존재할 때 $(x-p)^2 + (y-q)^2$를 최소화시키는 문제와 같다.\n",
"\n",
"따라서 라그랑지안 $\\mathcal{L}$은 다음과 같이 정의된다\n",
"\n",
"$$\n",
"\\mathcal{L}(x, y, \\lambda) = (x-p)^2 + (y-q)^2 + \\lambda(xlog2 - logy)\n",
"$$\n",
"\n",
"$\\nabla\\mathcal{L} = 0$이어야 하므로 다음 식이 성립한다\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\nabla_{x}\\mathcal{L} &= 2x - 2p + \\lambda log2 = 0 \\\\\n",
"\\nabla_{y}\\mathcal{L} &= 2y - 2q - \\frac{\\lambda}{y} = 0 \\\\\n",
"\\nabla_{\\lambda}\\mathcal{L} &= xlog2 - logy = 0\n",
"\\end{align}\n",
"$$\n",
"\n",
"1, 2번식을 정리하면 $x$와 $y$를 $\\lambda$에 대해서 표현할 수 있다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"x &= p - \\frac{\\lambda log2}{2} \\\\\n",
"y &= \\frac{q + \\sqrt{q^2 + 2\\lambda}}{2}(\\because y \\gt 0)\n",
"\\end{align}\n",
"$$\n",
"\n",
"이 식을 3번에 대입하면 $\\lambda$에 대한 식으로 바뀌고, 이는 newton raphson method로 해를 구할 수 있다. 이렇게 구한 좌표 $(x, y)$와 $(p, q)$간의 거리가 $y=2^x$와 $(p, q)$ 사이의 거리가 된다."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import math\n",
"from scipy import optimize\n",
"from functools import partial\n",
"log2 = math.log(2)\n",
"\n",
"def x(p, l):\n",
" return p - l*log2/2\n",
" \n",
"def y(q, l):\n",
" return (q + math.sqrt(q**2 + 2*l))/2\n",
"\n",
"def f(p, q, l):\n",
" return x(p, l)*log2 - math.log(y(q, l))\n",
"\n",
"def dist(p, q):\n",
" \"\"\"Get Euclidean distance between y=2^x and (p, q)\"\"\"\n",
" l = optimize.newton(lambda l: f(p, q, l), 1.0)\n",
" x_ = x(p, l)\n",
" y_ = 2**x_\n",
" return math.sqrt((x_-p)**2 + (y_-q)**2)\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2.7012892057857038e-15\n"
]
}
],
"source": [
"# test\n",
"print(dist(3, 8))"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"196"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"points = []\n",
"for a in range(1, 11):\n",
" for b in range(1, 101):\n",
" if 1 < dist(a, b) <= 2:\n",
" points.append((a, b))\n",
"len(points)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"IOPub data rate exceeded.\n",
"The notebook server will temporarily stop sending output\n",
"to the client in order to avoid crashing it.\n",
"To change this limit, set the config variable\n",
"`--NotebookApp.iopub_data_rate_limit`.\n"
]
}
],
"source": [
"from plotly.offline import init_notebook_mode\n",
"init_notebook_mode()"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
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]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Plot them\n",
"import plotly.offline as py\n",
"import plotly.graph_objs as go\n",
"import numpy as np\n",
"\n",
"circle_r1 = [{\n",
" \"type\": \"circle\",\n",
" \"xref\": \"x\",\n",
" \"yref\": \"y\",\n",
" \"x0\": px - 1.0,\n",
" \"x1\": px + 1.0,\n",
" \"y0\": py - 1.0,\n",
" \"y1\": py + 1.0,\n",
" \"line\": {\"color\": \"rgba(50, 171, 96, 0.3)\"}\n",
"} for px, py in points]\n",
"\n",
"circle_r2 = [{\n",
" \"type\": \"circle\",\n",
" \"xref\": \"x\",\n",
" \"yref\": \"y\",\n",
" \"x0\": px - 2.0,\n",
" \"x1\": px + 2.0,\n",
" \"y0\": py - 2.0,\n",
" \"y1\": py + 2.0,\n",
" \"line\": {\"color\": \"rgba(50, 171, 96, 0.3)\"}\n",
"} for px, py in points]\n",
"\n",
"curve_x = np.linspace(-1, 11, 200)\n",
"curve_y = np.power(2, curve_x)\n",
"\n",
"data = [go.Scatter(x=curve_x, y=curve_y)]\n",
"layout = {\n",
" \"height\": 3000,\n",
" \"xaxis\": {\n",
" \"range\": [-1, 11]\n",
" },\n",
" \"yaxis\": {\n",
" \"scaleanchor\": \"x\",\n",
" \"range\": [-1, 101]\n",
" },\n",
" \"shapes\": [\n",
" *circle_r1,\n",
" *circle_r2\n",
" ]\n",
"}\n",
"\n",
"py.iplot({\n",
" \"data\": data,\n",
" \"layout\": layout\n",
"}, filename=\"result\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.1"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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