For given \(a, b \in \mathbb{R}\) and \(c \in \mathbb{Z}\), find all function \(f: \mathbb{R} \to \mathbb{R}\) which is continuous at 0 and satisfies
\[
f(ax) = f(bx) + x^c \quad \forall x\in \mathbb{R}\setminus \{0\}.
\]

Consider a convex \((n+2)\)-gon. Let \(a_n\)​ denote the number of ways to add non-crossing chords to this polygon, including the case where no chords are added (i.e., \(a_0=a_1=1\) and \(a_2=3\)).

Find a recurrence relation for the sequence \(a_n​\) and determine its generating function.

Let \( X \) and \( Y \) be closed manifolds, and suppose \( X \) is a cover of \( Y \).  

Prove or disprove that the induced map on the first homology is injective.

There are \(n+1\) hats, each labeled with a number from \(1\) to \(n+1\), and \(n\) people. Each person is randomly assigned exactly one hat, and each hat is assigned to at most one person (i.e., the assignment is injective). A person can see all other assigned hats but cannot see their own hat and the unassigned hat. Each person must independently guess the number on their own hat.

If everyone correctly guesses their own hat’s number, they win; otherwise, they lose. They may discuss a strategy before the hats are assigned, but no communication is allowed afterward.

Determine a strategy that maximizes their probability of winning.

Let \( X \in \mathbb{R}^{n \times n} \) be a symmetric matrix with eigenvalues \( \lambda_i \) and orthonormal eigenvectors \( u_i \). The spectral decomposition gives \( X = \sum_{i=1}^n \lambda_i u_i u_i^\top \). For a function \( f : \mathbb{R} \to \mathbb{R} \), define \( f(X) := \sum_{i=1}^n f(\lambda_i) u_i u_i^\top \). Let \( X, Y \in \mathbb{R}^{n \times n} \) be symmetric. Is it always true that \( e^{X+Y} = e^X e^Y \)? If not, under what conditions does the equality hold?

We write \(tx = (tx_0,…,tx_5)\) for \(x=(x_0,…,x_5)\in \mathbb{R^{6}}\) and \(t\in \mathbb{R}\). Find all real multivariate polynomials \(P(x)\) in \(x\) satisfying the following properties:
(a) \(P(tx) = t^d P(x)\) for all \(t\in \mathbb{R}\) and \(x\in \mathbb{R}^{6}\), where \(0\leq d \leq 15\) is an integer;
(b) \(P(x) =0\) if \(x_i = x_j\) with \(i\neq j\).

Consider any sequence \( a_1,\dots, a_n \) of non-negative integers in \(\{0,1,\dots, m\}\). Prove that \[|\{ a_i+ a_j + (j-i): 1\leq i < j \leq n \}|\geq m \] when \(m= \lfloor \frac{1}{4} n^{2/3} \rfloor \).

A bonus problem: Can you find a function \(f(n)=\omega(n^{2/3})\) such that the above statement is true when \(m = f(n) \)? Is there such a function with \(f(n)= \Omega(n)\)? (You would still get full points without answering the bonus question.)

Let \( X \) and \( Y \) be closed manifolds, and suppose \( X \) is a finite-sheeted cover of \( Y \).  Prove or disprove that if \( Y \) has a nontrivial first Betti number, then \( X \) also has a nontrivial first Betti number.

Prove or disprove that every homomorphism \( \pi_1(X) \to \pi_1(X)\) can be realized as the induced homomorphism of a continuous map \(X \to X\).