From 40f4c510b6047d95d07f03c9915a53bbf84789e4 Mon Sep 17 00:00:00 2001
From: Mindaugas <76015221+mindaugl@users.noreply.github.com>
Date: Mon, 5 May 2025 15:14:56 +0800
Subject: [PATCH] Add solution for the Euler problem 190 (#12664)

* Add solution for the Euler project problem 164.

* Add solution for the Euler project problem 190.

* Delete project_euler/problem_164/sol1.py

* Delete project_euler/problem_164/__init__.py

* Update sol1.py

* Update sol1.py

* Update sol1.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

---------

Co-authored-by: Maxim Smolskiy <mithridatus@mail.ru>
Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
---
 project_euler/problem_190/__init__.py |  0
 project_euler/problem_190/sol1.py     | 48 +++++++++++++++++++++++++++
 2 files changed, 48 insertions(+)
 create mode 100644 project_euler/problem_190/__init__.py
 create mode 100644 project_euler/problem_190/sol1.py

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() = }")