HMMT 二月 2024 · 冲刺赛 · 第 7 题

HMMT February 2024 — Guts Round — Problem 7

[6] Positive integers a , b , and c have the property that a , b , and c end in 4 , 2 , and 9 , respectively. Compute the minimum possible value of a + b + c .

解析

[6] Positive integers a , b , and c have the property that a , b , and c end in 4 , 2 , and 9 , respectively. Compute the minimum possible value of a + b + c . Proposed by: Derek Liu Answer: 17 Solution: This minimum is attained when ( a, b, c ) = (2 , 2 , 13) . To show that we cannot do better, observe that a must be even, so c ends in 3 or 7 . If c ≥ 13 , since a and b are even, it’s clear (2 , 2 , 13) is c optimal. Otherwise, c = 3 or c = 7 , in which case b can end in 2 only when b ends in 8 . However, no eighth power ends in 4 , so we would need b ≥ 18 (and a ≥ 2 ), which makes the sum 2 + 18 + 3 = 23 larger than 17 .