HMMT 十一月 2013 · 冲刺赛 · 第 32 题

英文解析

32. [ 17 ] Suppose that x and y are chosen randomly and uniformly from (0 , 1). What is the probability
    ⌊√ ⌋
    ∑
    ∞2
    x 1 πthat is even? Hint: = .
    n =12
    y n 6
    √
    2 2
    π 24 − π x
    Answer: 1 − OR Note that for every positive integer n , the probability that b c = n
    24 24 y
    ( )
    1 1 1 1 1
    is just the area of the triangle formed between (0 , 0) , (1 , ) , (1 , ), which is just − .
    2 2 2 2
    n ( n +1) 2 n ( n +1)
    √
    Thus the probability that b c is odd isxy
    ( ) ( )
    ∞ ∞ ∞
    ∑ ∑ ∑
    1 1 1 1 1 1 1 − = + −
    2 2 2 2 2
    2 (2 k − 1) (2 k ) 2 (2 k − 1) (2 k ) (2 k )
    k =1 k =1 k =1
    ∞ ∞
    ∑ ∑
    1 1 1 1 = −
    2 2
    2 n 4 kn =1
    k =1
    2 2 2
    π π π = − = .
    12 24 24
    π2
    Thus our answer is just 1 − .
    24