Let $a$, $b$, $c$ and $d$ positive real numbers with $a > c$ and $b < d$. Assume that \[a + \sqrt{b} \ge c + \sqrt{d} \qquad \text{and} \qquad \sqrt{a} + b \le \sqrt{c} + d\] Prove that $a + b + c + d > 1$. Proposed by Victor Domínguez

We say that a permutation $(a_1, \dots, a_n)$ of $(1, 2, \dots, n)$ is good if the sums $a_1 + a_2 + \dots + a_i$ are all distinct modulo $n$. Prove that there exists a positive integer $n$ such that there are at least $2020$ good permutations of $(1, 2, \dots, n)$. Proposed by Ariel García

Let $n$ be a fixed positive integer. Oriol has $n$ cards, each of them with a $0$ written on one side and $1$ on the other. We place these cards in line, some face up and some face down (possibly all on the same side). We begin the following process consisting of $n$ steps: 1) At the first step, Oriol flips the first card 2) At the second step, Oriol flips the first card and second card . . . n) At the last step Oriol flips all the cards Let $s_0, s_1, s_2, \dots, s_n$ be the sum of the numbers seen in the cards at the beggining, after the first step, after the second step, $\dots$ after the last step, respectively. a) Find the greatest integer $k$ such that, no matter the initial card configuration, there exists at least $k$ distinct numbers between $s_0, s_1, \dots, s_n$. b) Find all positive integers $m$ such that, for each initial card configuration, there exists an index $r$ such that $s_r = m$. Proposed by Dorlir Ahmeti

Let $ABC$ be a triangle. Suppose that the perpendicular bisector of $BC$ meets the circle of diameter $AB$ at a point $D$ at the opposite side of $BC$ with respect to $A$, and meets the circle through $A, C, D$ again at $E$. Prove that $\angle ACE=\angle BCD$. Proposed by José Manuel Guerra and Victor Domínguez

A ladder is a non-decreasing sequence $a_1, a_2, \dots, a_{2020}$ of non-negative integers. Diego and Pablo play by turns with the ladder $1, 2, \dots, 2020$, starting with Diego. In each turn, the player replaces an entry $a_i$ by $a_i'<a_i$, with the condition that the sequence remains a ladder. The player who gets $(0, 0, \dots, 0)$ wins. Who has a winning strategy? Proposed by Violeta Hernández

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that \[f(f(x) - y) = f(xy) + f(x)f(-y)\]for any two real numbers $x, y$. Proposed by Pablo Valeriano