HMMT 十一月 2013 · GEN 赛 · 第 3 题

HMMT November 2013 — GEN Round — Problem 3

[ 4 ] A 24-hour digital clock shows times h : m : s , where h , m , and s are integers with 0 ≤ h ≤ 23, 0 ≤ m ≤ 59, and 0 ≤ s ≤ 59. How many times h : m : s satisfy h + m = s ?

解析

[ 4 ] A 24-hour digital clock shows times h : m : s , where h , m , and s are integers with 0 ≤ h ≤ 23, 0 ≤ m ≤ 59, and 0 ≤ s ≤ 59. How many times h : m : s satisfy h + m = s ? Answer: 1164 We are solving h + m = s in 0 ≤ s ≤ 59, 0 ≤ m ≤ 59, and 0 ≤ h ≤ 23. If s ≥ 24, each h corresponds to exactly 1 solution, so we get 24(59 − 23) = 24(36) in this case. If s ≤ 23, we want the number of nonnegative integer solutions to h + m ≤ 23, which by lattice point counting (or balls ( ) 23+2 and urns) is = (23 + 2)(23 + 1) / 2 = 25 · 12. Thus our total is 12(72 + 25) = 12(100 − 3) = 1164. 2