HMMT 二月 2006 · TEAM1 赛 · 第 14 题

英文解析

14. [40] A number n is called bummed out if there is exactly one ordered pair of positiveintegers ( x, y ) such that
    2 2
    b x /y c + b y /x c = n.
    Find all bummed out numbers.
    Answer: 2, 6, 8, 10
    Solution: Suppose n is bummed out. If ( a, b ) is one solution for ( x, y ) to the given
    2 2
    equation b x /y c + b y /x c = n, then ( b, a ) is another, so the unique solution ( a, b ) betterhave the property that a = b and n = 2 a ≥ 2. In particular, n is an even positiveinteger.
    Now, if n = 2 a ≥ 12, then setting x = a − 1 ≥ 5, y = a + 1 ≥ 7, we have
    ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ⌊ ⌋
    2 2
    x y 4 4 + = a − 3 + + a + 3 + = 2 a = n,
    y x a + 1 a − 1
    so n cannot be bummed out.
    2 2
    Moreover, b 1 / 2 c + b 2 / 1 c = 4, so 4 is not bummed out. The only possibilities left are
    2, 6, 8, and 10.
    To check these, note that
    ⌊ ⌋ ⌊ ⌋
    2 2 2 2 2
    x y x x y 3 x
    √
    n = + > − 2 + + + ≥ − 2 +
    y x 2 y 2 y x 3
    so 4
    √
    x < ( n + 2) < . 53( n + 2) ,43
    and similarly for y . So we only have to check x, y ≤ b . 53(10 + 2) c = 6:3
    x \ y 1 2 3 4 5 6
    1 2 4 9 16 25 36
    2 4 4 5 9 12 18
    3 9 5 6 7 9 13
    4 16 9 7 8 9 11
    5 25 12 9 9 10 11
    6 36 18 13 11 11 12