IMC 2008

Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that $f(x) - f(y)$ is rational for all reals $x$ and $y$ such that $x - y$ is rational.

Denote by $V$ the real vector space of all real polynomials in one variable, and let $P : V \to \mathbb{R}$ be a linear map. Suppose that for all $f, g \in V$ with $P(fg) = 0$ we have $P(f) = 0$ or $P(g) = 0$. Prove that there exist real numbers $x_0$, $c$ such that $P(f) = c f(x_0)$ for all $f \in V$.

Let $p$ be a polynomial with integer coefficients and let $a_1 < a_2 < \dots < a_k$ be integers.

a) Prove that there exists $a \in \mathbb{Z}$ such that $p(a_i)$ divides $p(a)$ for all $i = 1, 2, \dots, k$.

b) Does there exist an $a \in \mathbb{Z}$ such that the product $p(a_1) \cdot p(a_2) \cdot \dots \cdot p(a_k)$ divides $p(a)$?

We say a triple $(a_1, a_2, a_3)$ of nonnegative reals is better than another triple $(b_1, b_2, b_3)$ if two out of the three following inequalities $a_1 > b_1$, $a_2 > b_2$, $a_3 > b_3$ are satisfied. We call a triple $(x,y,z)$ special if $x$, $y$, $z$ are nonnegative and $x + y + z = 1$. Find all natural numbers $n$ for which there is a set $S$ of $n$ special triples such that for any given special triple we can find at least one better triple in $S$.

Does there exist a finite group $G$ with a normal subgroup $H$ such that $|\operatorname{Aut} H| > |\operatorname{Aut} G|$?

For a permutation $\sigma = (i_1, i_2, \dots, i_n)$ of $(1, 2, \dots, n)$ define $D(\sigma) = \sum\limits_{k=1}^{n} |i_k - k|$. Let $Q(n,d)$ be the number of permutations $\sigma$ of $(1, 2, \dots, n)$ with $d = D(\sigma)$. Prove that $Q(n,d)$ is even for $d \ge 2n$.

Let $n$, $k$ be positive integers and suppose that the polynomial $x^{2k} - x^k + 1$ divides $x^{2n} + x^n + 1$. Prove that $x^{2k} + x^k + 1$ divides $x^{2n} + x^n + 1$.

Two different ellipses are given. One focus of the first ellipse coincides with one focus of the second ellipse. Prove that the ellipses have at most two points in common.

Let $n$ be a positive integer. Prove that $2^{n-1}$ divides

Let $\mathbb{Z}[x]$ be the ring of polynomials with integer coefficients, and let $f(x), g(x) \in \mathbb{Z}[x]$ be nonconstant polynomials such that $g(x)$ divides $f(x)$ in $\mathbb{Z}[x]$. Prove that if the polynomial $f(x) - 2008$ has at least $81$ distinct integer roots, then the degree of $g(x)$ is greater than $5$.

Let $n$ be a positive integer, and consider the matrix $A = (a_{ij})_{1 \le i,j \le n}$, where

Prove that $|\det A| = k^2$ for some integer $k$.

Let $\mathcal{H}$ be an infinite-dimensional real Hilbert space, let $d > 0$, and suppose that $S$ is a set of points (not necessarily countable) in $\mathcal{H}$ such that the distance between any two distinct points in $S$ is equal to $d$. Show that there is a point $y \in \mathcal{H}$ such that

is an orthonormal system of vectors in $\mathcal{H}$.