HMMT 二月 2022 · 冲刺赛 · 第 22 题

HMMT February 2022 — Guts Round — Problem 22

[12] The function f ( x ) is of the form ax + bx + c for some integers a , b , and c . Given that { f (177 883) , f (348 710) , f (796 921) , f (858 522) } = { 1 324 754 875 645 , 1 782 225 466 694 , 1 984 194 627 862 , 4 388 794 883 485 } , compute a .

解析

[12] The function f ( x ) is of the form ax + bx + c for some integers a , b , and c . Given that { f (177 883) , f (348 710) , f (796 921) , f (858 522) } = { 1 324 754 875 645 , 1 782 225 466 694 , 1 984 194 627 862 , 4 388 794 883 485 } , compute a . Proposed by: Daniel Zhu Answer: 23 Solution: We first match the outputs to the inputs. To start, we observe that since a ≥ 0 (since the 12 answer to the problem is nonnegative), we must either have f (858522) ≈ 4 . 39 · 10 or f (177883) ≈ 12