HMMT 二月 2011 · TEAM1 赛 · 第 8 题

英文解析

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
   Solution: We will show how to find a in the first equation; other coefficients can be evaluated inthe same way. a is the probability that, beginning with | T − H | = 0, the value of | T − H | remains 01
   after using the optimal strategy for one round. This only happens when the first three coins are not 1
   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 ( ) .
   3 1 12
   (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