Each punch can be centered anywhere in the plane and removes all points within distance 1 from its center. What is the minimum number of punches needed to remove every point in the annulus between the circles of radius 7 and 10 (with the same center)? Describe your construction. The person with the smallest number of punches earns +4, and the next four best answers earn +3.

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.

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 \(X \to X\).

Let \( f(n) \) denote the number of possible sequences of length \( n \), where each term is either \(0, 1,\) or \(-1\), such that the product of every three consecutive numbers is nonnegative. Compute \( f(33)\).

Is it possible to arrange the numbers \(1, 2, 3, \ldots, 2024\) in a sequence such that the difference between any two adjacent numbers is greater than \(1\) but less than \(4\)?

Count the number of distinct matrices \( A \), where two matrices are considered identical if one can be obtained from the other by rearranging rows and columns, that have the following properties:

1. \( A \) is a \( 7 \times 7 \) matrix and every entry of \( A \) is \( 0 \) or \( 1 \).
2. Each row of \( A\) contains exactly 3 non-zero entries.
3. For any two distinct rows \( i\) and \( j\) of \( A\), there exists exactly one column \( k \) such that \( A_{ik} \neq 0 \) and \( A_{jk} \neq 0 \).

Let \(A\) be a \(16 \times 16\) matrix whose entries are either \(1\) or \(-1\). What is the maximum value of the determinant of \(A\)?

A complex number \(z \in S^1 \smallsetminus \{1\} \) is called a Knotennullstelle if there exists a Laurent polynomial \(p(t) \in \mathbb{Z} [t,t^{-1}]\) such that \(p(1) =\pm 1\) and \(p(z)=0\). Prove that the collection of all Knotennullstelle numbers is a discrete subset of \(\mathbb{C}\).

A permutation \(\phi \colon \{ 1,2, \ldots, n \} \to \{ 1,2, \ldots, n \}\) is called a well-mixed if \(\phi (\{1,2, \ldots, k \}) \neq \{1,2, \ldots, k \}\) for each \(k<n\). What is the number of well-mixed permutations of \(\{ 1,2, \ldots, 15 \}\)?