HMMT 二月 2017 · ALGNT 赛 · 第 2 题
HMMT February 2017 — ALGNT Round — Problem 2
题目详情
英文原题
- Find the value of
∑
a b c 1
2 3 5
1 ≤ a<b<c
(i.e. the sum of over all triples of positive integers ( a, b, c ) satisfying a < b < c )1
a b c
2 3 5
解析
英文解析
- Find the value of
∑
a b c 1
2 3 5
1 ≤ a<b<c
(i.e. the sum of over all triples of positive integers ( a, b, c ) satisfying a < b < c )1
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 . Thena 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