PUMaC 2023 · 团队赛 · 第 7 题

PUMaC 2023 — Team Round — Problem 7

Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers a, b, c , 3 2 respectively. They then compute the polynomial f ( x ) = x + px + qx + r with roots a, b, c . m If the expected value of the sum of the squares of the coefficients of f ( x ) is for relatively n prime positive integers m, n , find the remainder when m + n is divided by 1000.

解析

Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers a, b, c , 3 2 respectively. They then compute the polynomial f ( x ) = x + px + qx + r with roots a, b, c . m If the expected value of the sum of the squares of the coefficients of f ( x ) is for relatively n prime positive integers m, n , find the remainder when m + n is divided by 1000. Proposed by Sunay Joshi Answer: 551 2 2 2 The sum of the squares of the coefficients is 1+ p + q + r . By Vieta’s formulas, p = − ( a + b + c ), q = ab + bc + ca , and r = − abc . By independence, the expected value of the sum of the squares is therefore 2 2 2 3 1 + (3 ν + 6 μ ) + (3 ν + 6 νμ ) + ν , 2 where μ = E ( a ) and ν = E ( a ). It is easy to check that μ = 7 / 2 and ν = 91 / 6. Plugging these values into the above yields the fraction 1169335 / 216, so that our answer is 551. 3