Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent:

(1) There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\displaystyle\sum_{n=1}^{\infty} \frac{a_n}{c_n}$ and $\displaystyle\sum_{n=1}^{\infty} \frac{c_n}{b_n}$ both converge;

(2) $\displaystyle\sum_{n=1}^{\infty} \sqrt{\frac{a_n}{b_n}}$ converges.

(Proposed by Tomáš Bárta, Charles University, Prague)

Does there exist a field such that its multiplicative group is isomorphic to its additive group?

(Proposed by Alexandre Chapovalov, New York University, Abu Dhabi)

Determine all rational numbers $a$ for which the matrix $$\begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{pmatrix}$$ is the square of a matrix with all rational entries.

(Proposed by Daniël Kroes, University of California, San Diego)

Find all differentiable functions $f : (0, \infty) \to \mathbb{R}$ such that $$\tag{2} f(b) - f(a) = (b - a) f'\bigl( \sqrt{ab} \bigr) \quad \text{for all } a, b > 0.$$ (Proposed by Orif Ibrogimov, National University of Uzbekistan)

Let $p$ and $q$ be prime numbers with $p < q$. Suppose that in a convex polygon $P_1 P_2 \dots P_{pq}$ all angles are equal and the side lengths are distinct positive integers. Prove that $$P_1 P_2 + P_2 P_3 + \dots + P_k P_{k+1} \ge \frac{k^3 + k}{2}$$ holds for every integer $k$ with $1 \le k \le p$.

(Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Berlin)

Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exist $k$ nonzero vectors $v_1, \dots, v_k$ in $\mathbb{R}^n$ such that for every pair $i, j$ of indices with $|i - j| > 1$ the vectors $v_i$ and $v_j$ are orthogonal.

(Proposed by Alexey Balitskiy, Moscow Institute of Physics and Technology and M.I.T.)

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that $a_0 = 0$ and $$a_{n+1}^3 = a_n^2 - 8 \quad \text{for } n = 0, 1, 2, \dots$$ Prove that the following series is convergent: $$\tag{1} \sum_{n=0}^{\infty} |a_{n+1} - a_n|.$$ (Proposed by Orif Ibrogimov, National University of Uzbekistan)

Let $\Omega = \{ (x, y, z) \in \mathbb{Z}^3 : y + 1 \ge x \ge y \ge z \ge 0 \}$. A frog moves along the points of $\Omega$ by jumps of length 1. For every positive integer $n$, determine the number of paths the frog can take to reach $(n, n, n)$ starting from $(0, 0, 0)$ in exactly $3n$ jumps.

(Proposed by Fedor Petrov and Anatoly Vershik, St. Petersburg State University)

Determine all pairs $P(x)$, $Q(x)$ of complex polynomials with leading coefficient 1 such that $P(x)$ divides $Q(x)^2 + 1$ and $Q(x)$ divides $P(x)^2 + 1$.

(Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro)

For $R > 1$ let $D_R = \{ (a, b) \in \mathbb{Z}^2 : 0 < a^2 + b^2 < R \}$. Compute $$\lim_{R \to \infty} \sum_{(a,b) \in D_R} \frac{(-1)^{a+b}}{a^2 + b^2}.$$ (Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro)