HMMT 二月 2021 · ALGNT 赛 · 第 2 题

HMMT February 2021 — ALGNT Round — Problem 2

Compute the number of ordered pairs of integers ( a, b ) , with 2 ≤ a, b ≤ 2021, that satisfy the equation − 4 − 3 log a log ba ( ) ( ) b a a = b .

解析

Compute the number of ordered pairs of integers ( a, b ) , with 2 ≤ a, b ≤ 2021, that satisfy the equation − 4 − 3 log a log ba ( ) ( ) b a a = b . Proposed by: Vincent Bian Answer: 43 Solution: Taking log of both sides and simplifying tives a 2 − 4 log a = (log b ) − 3 log b. b a a 1 Plugging in x = log b and using log a = gives a b log b a 3 2 x − 3 x + 4 = 0 . 2 − 1 We can factor the polynomial as ( x − 2)( x − 2)( x + 1) , meaning b = a or b = a . The second case is impossible since both a and b are positive integers. So, we need only count the number of √ 2 1 < a, b ≤ 2021 for which b = a , which is b 2021 c − 1 = 43.