Now you can be top 5% at mathematics, according to that stupid puzzle

5-percent.py

from sage.all import *

"""
Solve a/(b+c) + b/(a+c) + c/(a+b) = N
Reference: https://ami.uni-eszterhazy.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf
"""

N = 6  # Fun fact: There is no solution when N is odd!

print(f"[+] Finding solution for: a/(b+c) + b/(a+c) + c/(a+b) = {N}")

# Build an elliptic curve and mapping.
E = EllipticCurve([0, 4 * N**2 + 12 * N - 3, 0, 32 * (N + 3), 0])
T = [E(t) for t in E.torsion_subgroup()]

print(f"[+] Elliptic curve: {latex(E)}")


def point_to_sol(x, y):
    a = 8 * (N + 3) - x + y
    b = 8 * (N + 3) - x - y
    c = 2 * (-4 * (N + 3) - x * (N + 2))

    if a < 0:
        a, b, c = -a, -b, -c

    g = GCD(GCD(a, b), c)

    return a // g, b // g, c // g


# Get one integral point of E.
P = E.integral_points()
# Generator must be inside the "egg".
for p in P:
    if p[0] < 0:
        break
print(f"[+] Found integral point: {p}")
cnt = 1
q = p

# Iterate until we find positive solution.
is_found = False
while True:
    print(f"[*] Trying {cnt} * P")

    for qq in [q] + [q + t for t in T]:
        a, b, c = point_to_sol(qq[0], qq[1])

        if (a > 0) and (b > 0) and (c > 0):
            print("[+] Found a, b, c!")
            is_found = True
            break

    if is_found:
        break

    q += p
    cnt += 1

print(f"[+] a = {a}")
print(f"[+] b = {b}")
print(f"[+] c = {c}")

# Check!
assert a / (b + c) + b / (c + a) + c / (b + a) == N