HMMT 二月 2025 · 冲刺赛 · 第 21 题

HMMT February 2025 — Guts Round — Problem 21

[12] Compute the unique 5-digit positive integer abcde such that a ̸ = 0, c ̸ = 0, and 2 abcde = ( ab + cde ) . 2 2 2

解析

[12] Compute the unique 5-digit positive integer abcde such that a ̸ = 0, c ̸ = 0, and 2 abcde = ( ab + cde ) . Proposed by: Rishabh Das Answer: 88209 2 Solution: Let ab = X and cde = Y . The original equation is equivalent to 1000 X + Y = ( X + Y ) . 2 Taking this modulo 999, we get ( X + Y ) ≡ X + Y (mod 999). Therefore, 27 · 37 = 999 divides ( X + Y )( X + Y − 1). Since gcd( X + Y, X + Y − 1) = 1, each of 27 and 37 can divide at most one of X + Y and X + Y − 1. Therefore, X + Y is either 0 or 1 modulo 27, as well as 0 or 1 modulo 37. By the Chinese Remainder Theorem, the four possible residues for X + Y modulo 999 are 0, 1, 297, 2 and 703. Since ( X + Y ) = abcde must be a 5-digit integer, we know 100 ≤ X + Y ≤ 316, so the only 2 2 possible value for X + Y is 297. Thus, the answer is ( X + Y ) = 297 = 88209 . 2 2 2