HMMT 二月 2019 · 冲刺赛 · 第 2 题

HMMT February 2019 — Guts Round — Problem 2

[ 3 ] There are 100 people in a room with ages 1 , 2 , . . . , 100 . A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room?

解析

[ 3 ] There are 100 people in a room with ages 1 , 2 , . . . , 100 . A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room? Proposed by: Yuan Yao Answer: 43 For a cute pair ( a, b ) we would have b a a ≥ + 7 , b ≥ + 7 . 2 2 Solving the system, we get that a and b must both be at least 14. However 14 could only be paired with itself or a smaller number; therefore, only people with age 15 or above can be paired with someone 100 − 14 of different age. Pairing consecutive numbers (15 , 16) , (17 , 18) , . . . , (99 , 100) works, giving = 43 2 pairs.