Let \(f(t)=(t^{pq}-1)(t-1) \) and \(g(t)=(t^{p}-1)(t^q-1) \) where \(p\) and \(q\) are relatively prime positive integers. Prove that \(\frac{f(t)}{g(t)}\) can be written as a polynomial where it has just \(1\) or \(-1\) as coefficients. (For example, when \(p=2\) and \(q=3\), we have that \(\frac{f(t)}{g(t)} = t^2-t+1\).)

Let \(p\) be a prime number at least three and let \(k\) be a positive integer smaller than \(p\). Given \(a_1,\dots, a_k\in \mathbb{F}_p\) and distinct elements \(b_1,\dots, b_k\in \mathbb{F}_p\), prove that there exists a permutation \(\sigma\) of \([k]\) such that the values of \(a_i + b_{\sigma(i)}\) are distinct modulo \(p\).

Let \(S\) be a set of distinct \(20\) integers. A set \(T_A\) is defined as \(T_A:=\{ s_1+s_2+s_3 \mid s_1, s_2, s_3 \in S\}\). What is the smallest possible cardinality of \(T_A\)?

Let \(\mathbb{S}_n\) be the set of all permutations of \([n]=\{1,\dots, n\}\). For positive real numbers \(d_1,\dots, d_n\), prove \[ \sum_{\sigma\in \mathbb{S}_n} \frac{1}{ d_{\sigma(1)}(d_{\sigma(1)}+d_{\sigma(2)}) \dots (d_{\sigma(1)}+\dots + d_{\sigma(n)}) } = \frac{1}{d_1\dots d_n}.\]

Find a pair of nonisomorphic nonabelian groups so that their abelianizations are isomorphic and their commutator subgroups are perfect.

Let \(\phi = \frac{1+\sqrt{5}}{2}\). Let \(f(1)=1\) and for \(n\geq 1\), let
\[ f(n+1) = \left\{\begin{array}{ll}
f(n)+2 & \text{ if } f(f(n)-n+1)=n \\
f(n)+1 & \text{ otherwise}.
\end{array}\right.\]
Prove that \(f(n) = \lfloor \phi n \rfloor\), and determine when \(f(f(n)-n+1)\neq n\) holds.

Let \(\{x_1, x_2, \ldots, x_{21}\} = \{-10, -9, \ldots, -1, 0, 1, \ldots, 9, 10\}\). What is the largest possible value of \(x_1x_2x_3+x_4x_5x_6+\cdots + x_{19}x_{20}x_{21}\)?