HMMT 十一月 2022 · 团队赛 · 第 5 题

HMMT November 2022 — Team Round — Problem 5

[40] A triple of positive integers ( a, b, c ) is tasty if lcm( a, b, c ) | a + b + c − 1 and a < b < c. Find the sum of a + b + c across all tasty triples.

解析

[40] A triple of positive integers ( a, b, c ) is tasty if lcm( a, b, c ) | a + b + c − 1 and a < b < c. Find the sum of a + b + c across all tasty triples. Proposed by: Isaac Zhu Answer: 44 Solution: The condition implies c | b + a − 1. WLOG assume c > b > a ; since b + a − 1 < 2 c we must have b + a − 1 = c . Substituting into b | a + c − 1 and a | c + b − 1 gives b | 2 a − 2 a | 2 b − 2 . Since 2 a − 2 < 2 b we must either have a = 1 (implying a = b , bad) or 2 a − 2 = b = ⇒ a | 4 a − 6 = ⇒ a = 2 , 3 , 6. If a = 2 then b = 2. Otherwise, if a = 3 we get (3 , 4 , 6) and if a = 6 we get (6 , 10 , 15), so answer is 13 + 31 = 44.