Alice and Bob play the following game with \( S=\{1,\dots, 777\} \).

Alice picks a number \(x \in S\) without telling anyone and Bob will guess what the number is at the end of the game. Alice is malicious so that she can always change her number \(x\) at any time until the end of the game.

In each round, Bob picks a subset \(T\subseteq S\) and asks a following question to Alice: “is your \(x\) belong to \(T\)?” Alice must say either Yes or No. At the end of the game, Bob guesses her \(x\) first and then Alice reveals her number \(x\) (Alice can still change her number after she listen to Bob’s guess and before revealing her number). According to her final number \(x\), each of her previous answers are determined to be either a truth or a lie.

Bob wins if Alice end up lying more than three times or his answer is correct. Alice wins if Bob’s answer is wrong and at most three of her answers are lies. Prove that if a game consists of twenty rounds, then no matter what Bob does Alice can always win.

Let \(S_g\) denote the closed orientable connected surface of genus \(g\). Suppose we glue triangles along the edges so that the resulting space is \(S_g\) and the intersection of any two triangles are either empty or a single edge. Let \( n(g) \) be the minimum number of triangles one needs to make \(S_g\) while satisfying the above rule. What are \( n(1), n(2), n(3) \)? Does the limit \( \lim_{g \to \infty} n(g)/g \) exist?

Consider the cards with labels \( 1,\dots, n \) in some order. If the top card has label \(m \), we reverse the order of the top \( m \) cards. The process stops only when the card with label \( 1\) is on the top. Prove that the process must stop in at most \( (1.7)^n \) steps.

Prove or disprove that a surjective homomorphism from a finitely generated abelian group to itself is an isomorphism.

Let \( m_0=n \). For each \( i\geq 0 \), choose a number \( x_i \) in \( \{1,\dots, m_i\} \) uniformly at random and let \( m_{i+1}= m_i – x_i\). This gives a random vector \( \mathbf{x}=(x_1,x_2, \dots) \). For each \( 1\leq k\leq n\), let \( X_k \) be the number of occurrences of \( k \) in the vector \( \mathbf{x} \).

For each \(1\leq k\leq n\), let \(Y_k\) be the number of cycles of length \(k\) in a permutation of \( \{1,\dots, n\} \) chosen uniformly at random. Prove that \( X_k \) and \(Y_k\) have the same distribution.

Say there are n points. For each pair of points, we add an edge with probability 1/3. Let \(P_n\) be the probability of the resulting graph to be connected (meaning any two vertices can be joined by an edge path). What can you say about the limit of \(P_n\) as n tends to infinity?

There are \(n\) people participating to a chess tournament and every two players play exactly one game against each other. The winner receives \(1\) point and the loser gets \(0\) point and if the game is a draw, each player receives \(0.5\) points. Prove that if at least \(3/4\) of the games are draws, then there are two players with the same total scores.