HMMT 十一月 2024 · GEN 赛 · 第 3 题

HMMT November 2024 — GEN Round — Problem 3

Suppose that a , b , and c are distinct positive integers such that a b = a . Across all possible values of a , b , and c , compute the minimum value of a + b + c .

解析

Suppose that a , b , and c are distinct positive integers such that a b = a . Across all possible values of a , b , and c , compute the minimum value of a + b + c . Proposed by: Derek Liu Answer: 13 Solution: We claim that (8 , 2 , 3) is the desired solution. c − b c Observe that a = b , so clearly a ̸ = 1 and b < a . Furthermore, a and b must be distinct powers of the same integer. 2 If a and b were powers of an integer n > 2, then we would have a + b + c ≥ 3 + 3 + 1 = 13. Thus, we only need to consider when they are powers of 2. c If ( a, b ) = (4 , 2) then ( c − b ) = , so c = 4, which makes the values not distinct. 2 If ( a, b ) = (8 , 2) we get our aforementioned solution. Any other ( a, b ) sum to at least 12, in which case a + b + c ≥ 13. Thus 13 is minimal.