HMMT 二月 2005 · CALC 赛 · 第 18 题

英文解析

18. What is f ( x ) f ( x ) dx ?
    Solution: 486/50
    ∫ ∫
    1 f (1)
    ′ 2 21
    0 = f ( x ) f ( x ) dx = u du = ( f (1) − f (0) ), and
    0 f (0)2
    ∫ ∫
    1 f (1)
    2 ′ 2 3 31
    18 = f ( x ) f ( x ) dx = u du = ( f (1) − f (0) ) .
    0 f (0)3
    The first equation implies f (0) = ± f (1). The second equation shows that f (0) 6 = f (1),
    3 3 3
    and in fact 54 = f (1) − f (0) = 2 f (1) , so f (1) = 3 and f (0) = − 3. Then
    ∫ ∫
    1 f (1)
    1 1 486
    4 ′ 4 5 5
    f ( x ) f ( x ) dx = u du = ( f (1) − f (0) ) = (243 + 243) = .
    5 5 5
    0 f (0)
    ′ 2 ′′