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