HMMT February, 2018

Combinatorics Difficulty 5.2 AIME, harder Prove it United States

Problem:

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?

Solution

Solution:

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, so the answer is $2^{10}-2=1022$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.