HMMT 二月 2004 · CALC 赛 · 第 10 题
HMMT February 2004 — CALC Round — Problem 10
题目详情
英文原题
- Let P ( x ) = x − x + x + . Let P ( x ) = P ( x ), and for n ≥ 1, let P ( x ) =
2 4
∫
[ n ] [2004]1
P ( P ( x )). Evaluate P ( x ) dx .
10
解析
英文解析
- Let P ( x ) = x − x + x + . Let P ( x ) = P ( x ), and for n ≥ 1, let P ( x ) =
2 4
∫
[ n ] [2004]1
P ( P ( x )). Evaluate P ( x ) dx .
Solution: 1 / 20
[ k ]
By Note that P (1 − x ) = 1 − P ( x ). It follows easily by induction that P (1 − x ) =
[ k ]
1 − P ( x ) for all positive integers k . Hence
∫ ∫
1 1
[2004] [2004]
P ( x ) dx = 1 − P (1 − x ) dx
0 0
∫
[2004]1 = 1 − P (1 − x ) dx
∫0
[2004]1 = 1 − P ( u ) du ( u = 1 − x ) .
∫0
[2004]1
Hence P ( x ) dx = 1 / 2.
30