PUMaC 2022 · 代数（B 组） · 第 1 题

PUMaC 2022 — Algebra (Division B) — Problem 1

Let q be the sum of the expressions a over all permutations ( a , a , a , a ) of (1 , 2 , 3 , 4). 1 2 3 4 1 Determine ⌊ q ⌋ .

解析

Let q be the sum of the expressions a over all permutations ( a , a , a , a ) of (1 , 2 , 3 , 4). 1 2 3 4 1 Determine ⌊ q ⌋ . Proposed by Frank Lu Answer: 8 We perform casework on the position of the 1. If a = 1, then we obtain a contribution of 1 1 1 1 1 3! · 1 = 6. If a = 1, then we obtain a contribution of 2! · ( + + ) = 2 + . If a = 1, then the 2 3 2 3 4 6 1 1 1 1 1 1 1 1 1 1 1 1 1 contribution is + + + + + , which is bounded by = + + + + + . 3 4 2 4 2 3 2 2 3 3 4 4 2 8 16 8 16 16 16 − 81 1 Finally, if a = 1, then the contribution is bounded by 6 · 2 , which is less than , say. 4 6 1 1 1 Therefore q ≥ 8 and q ≤ 8 + + + < 9. Our answer is thus ⌊ q ⌋ = 8. 6 2 6