HMMT 二月 2023 · ALGNT 赛 · 第 7 题
HMMT February 2023 — ALGNT Round — Problem 7
题目详情
- If a , b , c , and d are pairwise distinct positive integers that satisfy lcm( a, b, c, d ) < 1000 and a + b = c + d , compute the largest possible value of a + b .
解析
- If a , b , c , and d are pairwise distinct positive integers that satisfy lcm( a, b, c, d ) < 1000 and a + b = c + d , compute the largest possible value of a + b . Proposed by: Ankit Bisain Answer: 581 lcm( a,b,c,d ) ′ ′ ′ ′ ′ ′ ′ ′ Solution: Let a = . Define b , c , and d similarly. We have that a , b , c , and d are a pairwise distinct positive integers that satisfy 1 1 1 1
- = + . ′ ′ ′ ′ a b c d Let T be the above quantity. We have a + b = T lcm( a, b, c, d ) , 1 1 1 ′ ′ ′ ′ so we try to maximize T . Note that since + < , we cannot have any of a , b , c , and d be 1. At 2 3 1 most one of them can be 2, so at least one side of the equation must have both denominators at least