HMMT 二月 2025 · ALGNT 赛 · 第 2 题

HMMT February 2025 — ALGNT Round — Problem 2

Mark writes the expression abcd on the board, where abcd is a four-digit number and a ̸ = 0. Derek, √ a toddler, decides to move the a , changing Mark’s expression to a bcd . Surprisingly, these two expres- sions are equal. Compute the only possible four-digit number abcd .

解析

Mark writes the expression abcd on the board, where abcd is a four-digit number and a ̸ = 0. Derek, √ a toddler, decides to move the a , changing Mark’s expression to a bcd . Surprisingly, these two expres- sions are equal. Compute the only possible four-digit number abcd . Proposed by: Pitchayut Saengrungkongka Answer: 3375 √ √ Solution: Let x = bcd . Then, we rewrite the given condition abcd = a bcd as 2 1000 a + x = a x, which simplifies as 2 ( a − 1) x = 1000 a. 2 2 2 In particular, a − 1 divides 1000 a . Since gcd( a − 1 , a ) = 1, it follows that a − 1 | 1000. The only a ∈ { 1 , 2 , . . . , 9 } that satisfies this is a = 3. Then 8 x = 3000, so x = 375. Thus abcd = 3375 .