For each \( i \in \mathbb{N}\), let \(F_i\) be the \(i\)-th Fibonacci number where \(F_0=0, F_1=1\) and \(F_{i+1}=F_{i}+F_{i-1}\) for each \(i\geq 1\).
For \(n>m\), we divide \(F_n\) by \(F_m\) to obtain the remainder \(R\). Prove that either \(R\) or \(F_m-R\) is a Fibonacci number.
Suppose we choose a point on the unit circle in the plane at random with the uniform probability measure on the circle. When we choose n points in that way, what is the probability of the n-gon obtained as the convex hull of the chosen points has the area bigger than \( \pi/2 \) in terms of n?
Let \( S \) be the unit sphere in \( \mathbb{R}^n \), centered at the origin, and \( P_1 P_2 \dots P_{n+1} \) a regular simplex inscribed in \( S \). Prove that for a point \( P \) inside \( S \),
\[
\sum_{i=1}^{n+1} (PP_i)^4
\]
depends only on the distance \( OP \) (and \(n\)).
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?
Let \( n \) be a positive integer. Determine all continuous functions \(f: [0, 1] \to \mathbb{R}\) such that
\[
f(x_1) + \dots + f(x_n) =1
\]
for all \( x_1, \dots, x_n \in [0, 1] \) satisfying \( x_1 + \dots + x_n = 1\).
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 \( A \) be an \( n \times n \) Hermitian matrix and \( \lambda_1 (A) \geq \lambda_2 (A) \geq \dots \geq \lambda_n (A) \) the eigenvalues of \( A \). Prove that for any \( 1 \leq k \leq n \)
\[
A \mapsto \lambda_1 (A) + \lambda_2 (A) + \dots + \lambda_k (A)
\]
is a convex function.
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.