HMMT 二月 2023 · 冲刺赛 · 第 13 题

HMMT February 2023 — Guts Round — Problem 13

[14] Suppose a , b , c , and d are pairwise distinct positive perfect squares such that a = c . Compute the smallest possible value of a + b + c + d .

解析

[14] Suppose a , b , c , and d are pairwise distinct positive perfect squares such that a = c . Compute the smallest possible value of a + b + c + d . Proposed by: Luke Robitaille Answer: 305 Solution: Note that if a and c are divisible by more than one distinct prime, then we can just take x the prime powers of a specific prime. Thus, assume a and c are powers of a prime p . Assume a = 4 y and c = 4 . Then xb = yd . Because b and d are squares, the ratio of x to y is a square, so assume x = 1 and y = 4. We can’t take b = 4 and c = 1, but we instead can take b = 36 and c = 9. It can be checked that other values of x 36 9 and y are too big. This gives 4 = 256 , which gives a sum of 305. 4 If a and c are powers of 9, then max( a, c ) ≥ 9 , which is already too big. Thus, 305 is optimal.