HMMT 二月 2004 · CALC 赛 · 第 10 题

HMMT February 2004 — CALC Round — Problem 10

Let P ( x ) = x − x + x + . Let P ( x ) = P ( x ), and for n ≥ 1, let P ( x ) = 2 4 ∫ 1 [ n ] [2004] P ( P ( x )). Evaluate P ( x ) dx . 0 1

解析

Let P ( x ) = x − x + x + . Let P ( x ) = P ( x ), and for n ≥ 1, let P ( x ) = 2 4 ∫ 1 [ n ] [2004] P ( P ( x )). Evaluate P ( x ) dx . 0 Solution: 1 / 2 [ 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 ∫ 1 [2004] = 1 − P (1 − x ) dx 0 ∫ 1 [2004] = 1 − P ( u ) du ( u = 1 − x ) . 0 ∫ 1 [2004] Hence P ( x ) dx = 1 / 2. 0 3