HMMT 二月 2017 · ALGNT 赛 · 第 2 题

HMMT February 2017 — ALGNT Round — Problem 2

Find the value of ∑ 1 a b c 2 3 5 1 ≤ a<b<c 1 (i.e. the sum of over all triples of positive integers ( a, b, c ) satisfying a < b < c ) a b c 2 3 5

解析

Find the value of ∑ 1 a b c 2 3 5 1 ≤ a<b<c 1 (i.e. the sum of over all triples of positive integers ( a, b, c ) satisfying a < b < c ) a b c 2 3 5 Proposed by: Alexander Katz Answer: 1/1624 Let x = b − a and y = c − b so that b = a + x and c = a + x + y . Then a b c a a + x a + x + y a x y 2 3 5 = 2 3 5 = 30 15 5 and a, x, y are any positive integers. Thus ∑ ∑ 1 1 = a b c a x y 2 3 5 30 15 5 1 ≤ a ≤ b<c 1 ≤ a,x,y ∑ ∑ ∑ 1 1 1 = a x y 30 15 5 1 ≤ a 1 ≤ x 1 ≤ y 1 1 1 = · · 29 14 4 1 = 1624