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 \(T_A\)?
Find all pairs of prime numbers \( (p, q) \) such that \( pq \) divides \( p^p + q^q + 1 \).
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.
Suppose that \( f: [a, b] \to \mathbb{R} \) is a smooth, convex function, and there exists a constant \( t>0 \) such that \( f'(x) \geq t \) for all \( x \in (a, b) \). Prove that
\[
\left| \int_a^b e^{i f(x)} dx \right| \leq \frac{2}{t}.
\]
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 \(x_1x_2x_3+x_4x_5x_6+\cdots + x_{19}x_{20}x_{21}\)?
Find all integers \( n \) such that \( n^4 + n^3 + n^2 + n + 1 \) is a perfect square.
Determine the minimum number of hyperplanes in \(\mathbb{R}^n\) that do not contain the origin but they together cover all points in \(\{0,1\}^n\) except the origin.