Add solution for the Euler project problem 190.

Author: mindaugl

project_euler/problem_190/__init__.py

@@ -0,0 +1,50 @@
+"""
+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.
+
+>>> solution(5)
+10
+
+"""
+
+
+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
+
+    """
+
+    ans = 0
+    for m in range(2, n + 1):
+        x1 = 2 / (m + 1)
+        capital_p = 1.0
+        for i in range(1, m + 1):
+            xi = i * x1
+            capital_p *= pow(xi, i)
+        ans += int(capital_p)
+    return ans
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")