HMMT 二月 2024 · ALGNT 赛 · 第 4 题

HMMT February 2024 — ALGNT Round — Problem 4

Let f ( x ) be a quotient of two quadratic polynomials. Given that f ( n ) = n for all n ∈ { 1 , 2 , 3 , 4 , 5 } , compute f (0) .

解析

Let f ( x ) be a quotient of two quadratic polynomials. Given that f ( n ) = n for all n ∈ { 1 , 2 , 3 , 4 , 5 } , compute f (0) . Proposed by: Pitchayut Saengrungkongka 24 Answer: 17 3 Solution: Let f ( x ) = p ( x ) /q ( x ). Then, x q ( x ) − p ( x ) has 1 , 2 , 3 , 4 , 5 as roots. Therefore, WLOG, let 3 5 4 3 x q ( x ) − p ( x ) = ( x − 1)( x − 2)( x − 3)( x − 4)( x − 5) = x − 15 x + 85 x − . . . 2 Thus, q ( x ) = x − 15 x +85, so q (0) = 85. Plugging x = 0 in the above equation also gives − p (0) = − 120. 120 24 Hence, the answer is = . 85 17 Remark. From the solution above, it is not hard to see that the unique f that satisfies the problem is 2 225 x − 274 x + 120 f ( x ) = . 2 x − 15 x + 85