HMMT 二月 2024 · 冲刺赛 · 第 15 题

HMMT February 2024 — Guts Round — Problem 15

[9] Let a ⋆ b = ab − 2 . Compute the remainder when (((579 ⋆ 569) ⋆ 559) ⋆ · · · ⋆ 19) ⋆ 9 is divided by 100 .

解析

[9] Let a ⋆ b = ab − 2 . Compute the remainder when (((579 ⋆ 569) ⋆ 559) ⋆ · · · ⋆ 19) ⋆ 9 is divided by 100 . Proposed by: Rishabh Das Answer: 29 Solution: Note that (10 a + 9) ⋆ (10 b + 9) = (100 ab + 90 a + 90 b + 81) − 2 ≡ 90( a + b ) + 79 (mod 100) , so throughout our process all numbers will end in 9 , so we will just track the tens digit. Then the ”new operation” is a † b ≡ − ( a + b ) + 7 mod 10 , where a and b track the tens digits. Now ( a † b ) † c ≡ ( − ( a + b ) + 7) † c ≡ a + b − c mod 10 . Thus, our expression has tens digit congruent to − 0 + 1 − 2 + 3 − · · · − 54 + 55 − 56 − 57 + 7 ≡ − 28 − 57 + 7 ≡ 2 mod 10 , making the answer 29 .