IMC 2013

Let $A$ and $B$ be real symmetric matrices with all eigenvalues strictly greater than $1$. Let $\lambda$ be a real eigenvalue of matrix $AB$. Prove that $|\lambda| > 1$.

Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function. Suppose $f(0) = 0$. Prove that there exists $\xi \in (-\pi/2, \pi/2)$ such that

There are $2n$ students in a school ($n \in \mathbb{N}$, $n \ge 2$). Each week $n$ students go on a trip. After several trips the following condition was fulfilled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?

Let $n \ge 3$ and let $x_1, \dots, x_n$ be nonnegative real numbers. Define $A = \sum\limits_{i=1}^{n} x_i$, $B = \sum\limits_{i=1}^{n} x_i^2$ and $C = \sum\limits_{i=1}^{n} x_i^3$. Prove that

Does there exist a sequence $(a_n)$ of complex numbers such that for every positive integer $p$ we have that $\sum\limits_{n=1}^{\infty} a_n^p$ converges if and only if $p$ is not a prime?

Let $z$ be a complex number with $|z + 1| > 2$. Prove that $|z^3 + 1| > 1$.

Let $p$ and $q$ be relatively prime positive integers. Prove that

(Here $\lfloor x \rfloor$ denotes the integer part of $x$.)

Suppose that $v_1, \dots, v_d$ are unit vectors in $\mathbb{R}^d$. Prove that there exists a unit vector $u$ such that

for $i = 1, 2, \dots, d$.

(Here $\cdot$ denotes the usual scalar product on $\mathbb{R}^d$.)

Does there exist an infinite set $M$ consisting of positive integers such that for any $a, b \in M$, with $a < b$, the sum $a + b$ is square-free?

(A positive integer is called square-free if no perfect square greater than $1$ divides it.)

Consider a circular necklace with $2013$ beads. Each bead can be painted either white or green. A painting of the necklace is called good, if among any $21$ successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd.

(Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings.)