project_euler.problem_190.sol1
==============================

.. py:module:: project_euler.problem_190.sol1

.. autoapi-nested-parse::

   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.



Functions
---------

.. autoapisummary::

   project_euler.problem_190.sol1.solution


Module Contents
---------------

.. py:function:: 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