Problem

Source: India TST 2023 Day 3 P2

Tags: combinatorics, graph theory



In a school, every pair of students are either friends or strangers. Friendship is mutual, and no student is friends with themselves. A sequence of (not necessarily distinct) students $A_1, A_2, \dots, A_{2023}$ is called mischievous if $\bullet$ Total number of friends of $A_1$ is odd. $\bullet$ $A_i$ and $A_{i+1}$ are friends for $i=1, 2, \dots, 2022$. $\bullet$ Total number of friends of $A_{2023}$ is even. Prove that the total number of mischievous sequences is even.