Problem

Source: 2019 CSMO Grade 10 P5

Tags: combinatorics, Combinatorial Number Theory



Let $S=\{1928,1929,1930,\cdots,1949\}.$ We call one of $S$’s subset $M$ is a red subset, if the sum of any two different elements of $M$ isn’t divided by $4.$ Let $x,y$ be the number of the red subsets of $S$ with $4$ and $5$ elements,respectively. Determine which of $x,y$ is greater and prove that.