Let \(A\) be an 8 by 8 integral unimodular matrix. Moreover, assume that for each \( x \in \mathbb{Z}^8 \), we have \(x^{\top} A x \) is even. What is the possible number of positive eigenvalues for \(A\)?

Let \(\varphi(x)\) be the Euler’s totient function. Let \(S = \{a_1,\dots, a_n\}\) be a set of positive integers such that for any \(a_i\), all of its positive divisors are also in \(S\). Let \(A\) be the matrix with entries \(A_{i,j} = gcd(a_i,a_j)\) being the greatest common divisors of \(a_i\) and \(a_j\). Prove that \(\det(A) = \prod_{i=1}^{n} \varphi(a_i)\).

Let \(S\) be the set of all 4 by 4 integral positive-definite symmetric unimodular matrices. Define an equivalence relation \( \sim \) on \(S\) such that for any \( A,B \in S\), we have \(A \sim B\) if and only if \(PAP^\top = B\) for some integral unimodular matrix \(P\). Determine \(S ~/\sim \).

Let \(a(n)\) be the number of unordered factorizations of \(n\) into divisors larger than \(1\). Prove that \(\sum_{n=2}^{\infty} \frac{a(n)}{n^2} = 1\).

Let \(n, i\) be integers such that \(1 \leq i \leq n\). Each subset of \( \{ 1, 2, \ldots, n \} \) with \( i\) elements has the smallest number. We define \( \phi(n,i) \) to be the sum of these smallest numbers. Compute \[ \sum_{i=1}^n \phi(n,i).\]

Let \(S(n,k)\) be the Stirling number of the second kind that is the number of ways to partition a set of \(n\) objects into \(k\) non-empty subsets. Prove the following equality \[ \det\left( \begin{matrix} S(m+1,1) & S(m+1,2) & \cdots & S(m+1,n) \\
S(m+2,1) & S(m+2,2) & \cdots & S(m+2,n) \\
\cdots & \cdots & \cdots & \cdots \\
S(m+n,1) & S(m+n,2) & \cdots & S(m+n,n) \end{matrix} \right) = (n!)^m \]

For a positive integer \(n\), let \(B\) and \(C\) be real-valued \(n\) by \(n\) matrices and \(O\) be the \(n\) by \(n\) zero matrix. Assume further that \(B\) is invertible and \(C\) is symmetric. Define \[A := \begin{pmatrix} O & B \\ B^T & C \end{pmatrix}.\] What is the possible number of positive eigenvalues for \(A\)?