HMMT 十一月 2019 · 冲刺赛 · 第 14 题

HMMT November 2019 — Guts Round — Problem 14

[9] Compute the sum of all positive integers n for which √ √ √ 9 n + 4 n + 2 − 3 n + 16 is an integer.

解析

[9] Compute the sum of all positive integers n for which √ √ √ 9 n + 4 n + 2 − 3 n + 16 is an integer. Proposed by: Milan Haiman Answer: 18 For the expression to be an integer at least one of n and n + 2 must be a perfect square. We also note that at most one of n and n + 2 can be a square, so exactly one of them is a square. √ √ Case 1: n is a perfect square. By our previous observation, it must be that 4 n + 2 = 3 n + 16 ⇒ n = 16. √ √ Case 2: n + 2 is a perfect square. By our previous observation, it must be that 9 n = 3 n + 16 ⇒ n = 2. Consequently, the answer is 16 + 2 = 18.