IMC 2018

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 $\sum\limits_{n=1}^{\infty} \dfrac{a_n}{c_n}$ and $\sum\limits_{n=1}^{\infty} \dfrac{c_n}{b_n}$ both converge;

(2) $\sum\limits_{n=1}^{\infty} \sqrt{\dfrac{a_n}{b_n}}$ converges.

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

Determine all rational numbers $a$ for which the matrix

is the square of a matrix with all rational entries.

Find all differentiable functions $f \colon (0, \infty) \to \mathbb{R}$ such that

Let $p$ and $q$ be prime numbers with $p < q$. Suppose that in a convex polygon $P_1P_2 \ldots P_{pq}$ all angles are equal and the side lengths are distinct positive integers. Prove that

holds for every integer $k$ with $1 \le k \le p$.

Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exist $k$ nonzero vectors $v_1, \ldots, v_k$ 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.

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that $a_0 = 0$ and

Prove that the following series is convergent:

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.

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 $P(x)^2 + 1$.

For $R > 1$ let $\mathcal{D}_R = \{(a, b) \in \mathbb{Z}^2 : 0 < a^2 + b^2 < R\}$. Compute