HMMT 二月 2016 · 冲刺赛 · 第 2 题
HMMT February 2016 — Guts Round — Problem 2
题目详情
英文原题
- [ 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 noticingit (and hence missing the train). What is the probability that Sherry catches the train in the next fiveminutes?
解析
英文解析
- This implies that for some j , 2 ≡ − d (mod 2 − 1). But notice that the powers of 2 (mod 2 − 1)
d − 1 dare 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 orderd − 1 d d − 1
for this to be true, we must have 2 + d ≥ 2 − 1 ⇐⇒ d + 1 ≥ 2 . This inequality is only truedfor 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 formj 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.