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project_euler/problem_190/sol1.py
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Add solution for the Euler problem 190 (#12664)
05 май 2025, 10:14
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05 май 2025, 10:14
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""" Project Euler Problem 190: https://projecteuler.net/problem=190 Maximising a Weighted Product Let S_m = (x_1, x_2, ..., x_m) be the m-tuple of positive real numbers with x_1 + x_2 + ... + x_m = m for which P_m = x_1 * x_2^2 * ... * x_m^m is maximised. For example, it can be verified that |_ P_10 _| = 4112 (|_ _| is the integer part function). Find Sum_{m=2}^15 = |_ P_m _|. Solution: - Fix x_1 = m - x_2 - ... - x_m. - Calculate partial derivatives of P_m wrt the x_2, ..., x_m. This gives that x_2 = 2 * x_1, x_3 = 3 * x_1, ..., x_m = m * x_1. - Calculate partial second order derivatives of P_m wrt the x_2, ..., x_m. By plugging in the values from the previous step, can verify that solution is maximum. """ def solution(n: int = 15) -> int: """ Calculate sum of |_ P_m _| for m from 2 to n. >>> solution(2) 1 >>> solution(3) 2 >>> solution(4) 4 >>> solution(5) 10 """ total = 0 for m in range(2, n + 1): x1 = 2 / (m + 1) p = 1.0 for i in range(1, m + 1): xi = i * x1 p *= xi**i total += int(p) return total if __name__ == "__main__": print(f"{solution() = }")