HMMT 二月 2002 · 冲刺赛 · 第 47 题

HMMT February 2002 — Guts Round — Problem 47

[9] The real function f has the property that, whenever a, b, n are positive integers such n 2 that a + b = 2 , the equation f ( a ) + f ( b ) = n holds. What is f (2002)?

解析

The real function f has the property that, whenever a, b, n are positive integers such n 2 that a + b = 2 , the equation f ( a ) + f ( b ) = n holds. What is f (2002)? 2 n n Solution: We know f ( a ) = n − f (2 − a ) for any a, n with 2 > a ; repeated application gives 2 2 2 2 2 2 f (2002) = 11 − f (46) = 11 − (6 − f (18)) = 11 − (6 − (5 − f (14))) 2 2 2 2 = 11 − (6 − (5 − (4 − f (2)))) . 2 2 2 2 2 But f (2) = 2 − f (2), giving f (2) = 2, so the above simplifies to 11 − (6 − (5 − (4 − 2))) = 96 .