HMMT 二月 2002 · 冲刺赛 · 第 47 题
HMMT February 2002 — Guts Round — Problem 47
题目详情
英文原题
- [9] The real function f has the property that, whenever a, b, n are positive integers suchn 2
that a + b = 2 , the equation f ( a ) + f ( b ) = n holds. What is f (2002)?
解析
英文解析
- The real function f has the property that, whenever a, b, n are positive integers suchn 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 applicationgives
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 .