HMMT 二月 2018 · 冲刺赛 · 第 3 题

HMMT February 2018 — Guts Round — Problem 3

[ 4 ] Allen and Yang want to share the numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10. How many ways are there to split all ten numbers among Allen and Yang so that each person gets at least one number, and either Allen’s numbers or Yang’s numbers sum to an even number?

解析

[ 4 ] Allen and Yang want to share the numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10. How many ways are there to split all ten numbers among Allen and Yang so that each person gets at least one number, and either Allen’s numbers or Yang’s numbers sum to an even number? Proposed by: Kevin Sun Answer: 1022 Since the sum of all of the numbers is odd, exactly one of Allen’s sum and Yang’s sum must be odd. Therefore any way of splitting the numbers up where each person receives at least one number is valid, 10 so the answer is 2 − 2 = 1022.