Let $\mathcal{F}$ be the family of bijective increasing functions $f\colon [0,1] \to [0,1]$, and let $a \in (0,1)$. Determine the best constants $m_a$ and $M_a$, such that for all $f \in \mathcal{F}$ we have \[m_a \leq f(a) + f^{-1}(a) \leq M_a.\] (Dan Schwarz)

Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. i) Prove at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle. ii) Assuming the three points inside the rectangle are three corners of it, prove at least two of the three concave quadrilaterals formed by these four points have perimeters lesser than that of the rectangle. (Dan Schwarz)

Consider the sequence $(a^n + 1)_{n\geq 1}$, with $a>1$ a fixed integer. i) Prove there exist infinitely many primes, each dividing some term of the sequence. ii) Prove there exist infinitely many primes, none dividing any term of the sequence. (Dan Schwarz)

Given a (fixed) positive integer $N$, solve the functional equation \[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) = f(k), \ \textrm{for all } k \in \mathbb{Z}.\] (Dan Schwarz)

Prove that for any integers $a,b$, the equation $2abx^4 - a^2x^2 - b^2 - 1 = 0$ has no integer roots. (Dan Schwarz)

Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. Prove that at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle. (Dan Schwarz)

Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$. i) Prove there exist infinitely many primes, none dividing any term of the sequence. ii) Prove there exist infinitely many primes, each dividing some term of the sequence. (Dan Schwarz)

A set $S$ of unit cells of an $n\times n$ array, $n\geq 2$, is said full if each row and each column of the array contain at least one element of $S$, but which has this property no more when any of its elements is removed. A full set having maximum cardinality is said fat, while a full set of minimum cardinality is said meagre. i) Determine the cardinality $m(n)$ of the meagre sets, describe all meagre sets and give their count. ii) Determine the cardinality $M(n)$ of the fat sets, describe all fat sets and give their count. (Dan Schwarz)