PUMaC 2008 · 团队赛 · 第 8 题

PUMaC 2008 — Team Round — Problem 8

(5 points) Suppose that the roots of the quadratic x + ax + b are α and β . Then α and β are 2 the roots of some quadratic x + cx + d. Find c in terms of a and b . 1 Team B

解析

Alice and Bob play a game with a coin. The coin is thrown and Alice wins $1 from Bob if it lands heads, and Bob $1 from Alice if it lands tails. Initially, Alice has 2008 dollars and Bob 2009. Find the sum of the probability that Alice wins and the expected number of tosses until one of them wins, expressed as a mixed number. 2 Team 2008 ( ANS: + 4034072. Let f ( n ) denote the probability that Alice wins if she has n dollars, and 4017 f ( n − 1)+ f ( n +1) let g ( n ) denote the expected time until the game ends. Then f (0) = 0, f ( n ) = , and 2 f (4017) = 1. Unwinding the algebraic identity, we have f ( n + 2) − f ( n + 1) = f ( n + 1) − f ( n ), so differences are constant and thus you have a linear function, and so the answer for the first part is g ( n +1)+ g ( n − 1) 2008 . g satisfies g (0) = g (4017) = 0 and g ( n ) = + 1. Thus, 4017 2 g ( n + 1) − g ( n ) = g ( n ) − g ( n − 1) − 2, and so, by successive differences, g is a quadratic with leading term − n 2 and roots at 0 and 4017, so g ( n ) = n (4017 − n ). CB: AL) 2 3 3