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 $5$.

Let $n \ge 2$ be an integer. What is the minimal and maximal possible rank of an $n \times n$ matrix whose $n^2$ entries are precisely the numbers $1, 2, \dots, n^2$?

Call a polynomial $P(x_1, \dots, x_k)$ good if there exist $2 \times 2$ real matrices $A_1, \dots, A_k$ such that

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 $n$ be a positive integer and $a_1, \dots, a_n$ be arbitrary integers. Suppose that a function $f : \mathbb{Z} \to \mathbb{R}$ satisfies $\sum\limits_{i=1}^{n} f(k + a_i \ell) = 0$ whenever $k$ and $\ell$ are integers and $\ell \ne 0$. Prove that $f = 0$.

How many nonzero coefficients can a polynomial $P(z)$ have if its coefficients are integers and $|P(z)| \le 2$ for any complex number $z$ of unit length?

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 $x$, $y$, and $z$ be integers such that $S = x^4 + y^4 + z^4$ is divisible by $29$. Show that $S$ is divisible by $29^4$.

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 $x \le y$.)

Let $n > 1$ be an odd positive integer and $A = (a_{ij})_{i,j=1\dots n}$ be the $n \times n$ matrix with

Find $\det A$.

For each positive integer $k$, find the smallest number $n_k$ for which there exist real $n_k \times n_k$ matrices $A_1, A_2, \dots, A_k$ such that all of the following conditions hold:

(1) $A_1^2 = A_2^2 = \dots = A_k^2 = 0$,

(2) $A_i A_j = A_j A_i$ for all $1 \le i, j \le k$, and

(3) $A_1 A_2 \dots A_k \ne 0$.

Let $f \ne 0$ be a polynomial with real coefficients. Define the sequence $f_0, f_1, f_2, \dots$ of polynomials by $f_0 = f$ and $f_{n+1} = f_n + f_n'$ for every $n \ge 0$. Prove that there exists a number $N$ such that for every $n \ge N$, all roots of $f_n$ are real.