HMMT 二月 2011 · TEAM1 赛 · 第 8 题

HMMT February 2011 — TEAM1 Round — Problem 8

(a) [ 5 ] Find a , a , and a , so that the following equation holds for all m ≥ 1: 1 2 3 p (0) = a p (0) + a p (2) + a p (4) m 1 m − 1 2 m − 1 3 m − 1 (b) [ 5 ] Find b , b , b , and b , so that the following equation holds for all m ≥ 1: 1 2 3 4 p (2) = b p (0) + b p (2) + b p (4) + b p (6) m 1 m − 1 2 m − 1 3 m − 1 4 m − 1 (c) [ 5 ] Find c , c , c , and c , so that the following equation holds for all m ≥ 1 and j ≥ 2: 1 2 3 4 p (2 j ) = c p (2 j − 2) + c p (2 j ) + c p (2 j + 2) + c p (2 j + 4) m 1 m − 1 2 m − 1 3 m − 1 4 m − 1

解析

(a) [ 5 ] Find a , a , and a , so that the following equation holds for all m ≥ 1: 1 2 3 p (0) = a p (0) + a p (2) + a p (4) m 1 m − 1 2 m − 1 3 m − 1 (b) [ 5 ] Find b , b , b , and b , so that the following equation holds for all m ≥ 1: 1 2 3 4 p (2) = b p (0) + b p (2) + b p (4) + b p (6) m 1 m − 1 2 m − 1 3 m − 1 4 m − 1 (c) [ 5 ] Find c , c , c , and c , so that the following equation holds for all m ≥ 1 and j ≥ 2: 1 2 3 4 p (2 j ) = c p (2 j − 2) + c p (2 j ) + c p (2 j + 2) + c p (2 j + 4) m 1 m − 1 2 m − 1 3 m − 1 4 m − 1 Solution: We will show how to find a in the first equation; other coefficients can be evaluated in 1 the same way. a is the probability that, beginning with | T − H | = 0, the value of | T − H | remains 0 1 after using the optimal strategy for one round. This only happens when the first three coins are not 3 1 all heads or all tails. Therefore, a = . It follows that b = . The rest of the terms are the binomial 1 1 4 4 1+1 3 coefficients of the expansion of ( ) . 2 3 1 1 (a) a = , a = , a = . 1 2 3 4 2 8 1 3 3 1 (b) b = , b = , b = , b = . 1 2 3 4 4 8 8 8 1 3 3 1 (c) c = , c = , c = , c = . 1 2 3 4 8 8 8 8