IMC 2007

Let $f$ be a polynomial of degree 2 with integer coefficients. Suppose that $f(k)$ is divisible by 5 for every integer $k$ . Prove that all coefficients of $f$ are divisible by 5.

Call a polynomial $P(x_{1}, \ldots, x_{k})$ good if there exist $2\times 2$ real matrices $A_{1}, \ldots, A_{k}$ such that $P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$

Find all values of $k$ for which all homogeneous polynomials with $k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

Let $G$ be a finite group. For arbitrary sets $U, V, W \subset G$ , denote by $N_{UVW}$ the number of triples $(x, y, z) \in U \times V \times W$ for which $xyz$ is the unity . Suppose that $G$ is partitioned into three sets $A, B$ and $C$ (i.e. sets $A, B, C$ are pairwise disjoint and $G = A \cup B \cup C$ ). Prove that $N_{ABC}= N_{CBA}.$

Let $f : \mathbb{R}\to \mathbb{R}$ be a continuous function. Suppose that for any $c > 0$ , the graph of $f$ can be moved to the graph of $cf$ using only a translation or a rotation. Does this imply that $f(x) = ax+b$ for some real numbers $a$ and $b$ ?

Let $C$ be a nonempty closed bounded subset of the real line and $f: C\to C$ be a nondecreasing continuous function. Show that there exists a point $p\in C$ such that $f(p) = p$ . (A set is closed if its complement is a union of open intervals. A function $g$ is nondecreasing if $g(x)\le g(y)$ for all $x\le y$ .)