HMMT 二月 2016 · 冲刺赛 · 第 2 题

HMMT February 2016 — Guts Round — Problem 2

[ 5 ] Sherry is waiting for a train. Every minute, there is a 75% chance that a train will arrive. However, she is engrossed in her game of sudoku, so even if a train arrives she has a 75% chance of not noticing it (and hence missing the train). What is the probability that Sherry catches the train in the next five minutes?

解析

This implies that for some j , 2 ≡ − d (mod 2 − 1). But notice that the powers of 2 (mod 2 − 1) d − 1 d are 1 , 2 , 4 , . . . , 2 (2 ≡ 1 so the cycle repeats). j d In order for the residues to match, we need 2 + d = c (2 − 1), where 0 ≤ j ≤ d − 1 and c ≥ 1. In order d − 1 d d − 1 for this to be true, we must have 2 + d ≥ 2 − 1 ⇐⇒ d + 1 ≥ 2 . This inequality is only true d for d = 1 , 2 , 3. We plug each of these into the original expression (2 − 1) n − d . j j For d = 1: n − 1 is a power of 2. This yields the set of solutions (2 + 2 , 2 + 1) for j ≥ 0. For d = 2: 3 n − 2 is a power of 2. Note that powers of 2 are − 2 (mod 3) if and only if it is an even 2 j 2 j 2 j 2 +2 2 +8 2 +2 power, so n = . This yields the solution set ( , ) , j ≥ 0. 3 3 3 For d = 3: 7 n − 3 is a power of 2. Powers of 2 have a period of 3 when taken (mod 7), so inspection 3 j +2 3 j +2 2 +24 2 +3 3 j +2 tells us 7 n − 3 = 2 , yielding the solution set ( , ) , j ≥ 0. 7 7 Therefore, all the solutions are of the form j j j ( m, n ) = (0 , 2 ) , (2 + 2 , 2 + 1) 2 j 2 j 3 j +2 3 j +2 2 + 8 2 + 2 2 + 24 2 + 3 ( , ) , ( , ) 3 3 7 7 for j ≥ 0. Restricting this family to m, n ≤ 100 gives 7 + 7 + 5 + 3 = 22.