HMMT 二月 2002 · CALC 赛 · 第 10 题

HMMT February 2002 — CALC Round — Problem 10

A continuous real function f satisfies the identity f (2 x ) = 3 f ( x ) for all x . If f ( x ) dx = 0 ∫ 2 1, what is f ( x ) dx ? 1 1

解析

A continuous real function f satisfies the identity f (2 x ) = 3 f ( x ) for all x . If ∫ ∫ 1 2 f ( x ) dx = 1, what is f ( x ) dx ? 0 1 ∫ ∫ 2 1 Solution: 5 Let S = f ( x ) dx . By setting u = 2 x , we see that f ( x ) dx = 1 1 / 2 ∫ ∫ ∫ 1 2 1 / 2 f (2 x ) / 3 dx = f ( u ) / 6 du = S/ 6. Similarly, f ( x ) dx = S/ 36, and in general 1 / 2 1 1 / 4 ∫ ∫ n − 1 1 / 2 1 n f ( x ) dx = S/ 6 . Adding finitely many of these, we have f ( x ) dx = S/ 6 + S/ 36 + n n 1 / 2 1 / 2 ∫ 1 n n · · · + S/ 6 = S · (1 − 1 / 6 ) / 5. Taking the limit as n → ∞ , we have f ( x ) dx = S/ 5. Thus 0 S = 5, the answer. 2