diff --git a/project_euler/problem_190/__init__.py b/project_euler/problem_190/__init__.py new file mode 100644 index 000000000..e69de29bb diff --git a/project_euler/problem_190/sol1.py b/project_euler/problem_190/sol1.py new file mode 100644 index 000000000..b18d45be1 --- /dev/null +++ b/project_euler/problem_190/sol1.py @@ -0,0 +1,48 @@ +""" +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() = }")