PUMaC 2015 · 代数（B 组） · 第 3 题

PUMaC 2015 — Algebra (Division B) — Problem 3

[ 4 ] Andrew and Blair are bored in class and decide to play a game. They pick a pair ( a, b ) with 1 ≤ a, b ≤ 100. Andrew says the next number in the geometric series that begins with a, b and Blair says the next number in the arithmetic series that begins with a, b . For how many pairs ( a, b ) is Andrew’s number minus Blair’s number a positive perfect square? 2 3 2

解析

[ 4 ] Andrew and Blair are bored in class and decide to play a game. They pick a pair ( a, b ) with 1 ≤ a, b ≤ 100. Andrew says the next number in the geometric series that begins with a, b and Blair says the next number in the arithmetic series that begins with a, b . For how many pairs ( a, b ) is Andrew’s number minus Blair’s number a positive perfect square? Solution: 2 2 2 b b − 2 ab + a Andrew will say and Blair will say 2 b − a and hence the difference will be = a a 2 ( b − a ) 2 . In order for this to be a perfect square, a must be a perfect square and a | ( b − a ) a √ √ 2 so a | b = ⇒ a | b . Since 1 ≤ a ≤ 100, 1 ≤ a ≤ 10 and the possible choices for b are √ √ √ √ √ 2 a, 3 a, . . . , b 100 / a c a or a total of b 100 / a c − 1 possibilities. Note that b 6 = a since the difference must be positive. So our answer is: ⌊ ⌋ 10 ∑ 100 − 1 = 281 i i =1 Author: Roy Zhao 1 2 3 2