HMMT 十一月 2013 · 冲刺赛 · 第 32 题

HMMT November 2013 — Guts Round — Problem 32

[ 17 ] Suppose that x and y are chosen randomly and uniformly from (0 , 1). What is the probability ⌊√ ⌋ ∑ 2 ∞ x 1 π that is even? Hint: = . 2 n =1 y n 6

解析

[ 17 ] Suppose that x and y are chosen randomly and uniformly from (0 , 1). What is the probability ⌊√ ⌋ ∑ 2 ∞ x 1 π that is even? Hint: = . 2 n =1 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) √ x Thus the probability that b c is odd is y ( ) ( ) ∞ ∞ ∞ ∑ ∑ ∑ 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 k n =1 k =1 2 2 2 π π π = − = . 12 24 24 2 π Thus our answer is just 1 − . 24