A problem from IOI 2014
Competitive programming problems turn out to be helpful for math olympiad training (and it’s fun too)! Let us look at IOI 2014 Problem 3: Let $n \ge 3$ be an integer. Alice and Bob play a game based on a graph of $n$ points, each labeled from $0$ to $n-1$. The graph is initially empty. In the $i^{\text{th}}$ round of the game, the following occurs in order: Alice picks a pair of integers $(u,v)$ such that $0\le u < v \le n-1$ that she didn't pick before, then Bob chooses whether $\{u, v\}$ should be an edge or not. If Bob chooses $\{u, v\}$ to be an edge, then the edge connecting $\{u, v\}$ is added to the graph. (Note that there are exactly $\binom{n}{2}$ rounds.) Alice wins if, before the last round, she can guarantee whether the graph is connected or not. Bob wins otherwise. Help Bob win the game. ...