SCHWEITZER 2003

Let $(X, <)$ be an arbitrary ordered set. Show that the elements of $X$ can be coloured by two colours in such a way that between any two points of the same colour there is a point of the opposite colour.

(translated by L. Erdős)

Let $p$ be a prime and let $M$ be an $n\times m$ matrix with integer entries such that $Mv\not\equiv 0\pmod{p}$ for any column vector $v\neq 0$ whose entries are $0$ are $1$ . Show that there exists a row vector $x$ with integer entries such that no entry of $xM$ is $0\pmod{p}$ .

(translated by L. Erdős)

Let $Z=\{ z_1,\ldots, z_{n-1}\}$ , $n\ge 2$ , be a set of different complex numbers such that $Z$ contains the conjugate of any its element. a) Show that there exists a constant $C$ , depending on $Z$ , such that for any $\varepsilon\in (0,1)$ there exists an algebraic integer $x_0$ of degree $n$ , whose algebraic conjugates $x_1, x_2, \ldots, x_{n-1}$ satisfy $|x_1-z_1|\le \varepsilon, \ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$ and $|x_0|\le \frac{C}{\varepsilon}$ . b) Show that there exists a set $Z=\{ z_1, \ldots, z_{n-1}\}$ and a positive number $c_n$ such that for any algebraic integer $x_0$ of degree $n$ , whose algebraic conjugates satisfy $|x_1-z_1|\le \varepsilon,\ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$ , it also holds that $|x_0|>\frac{c_n}{\varepsilon}$ .

(translated by L. Erdős)

Let $\{a_{n,1},\ldots, a_{n,n} \}_{n=1}^{\infty}$ integers such that $a_{n,i}\neq a_{n,j}$ for $1\le i<j\le n , n=2,3,\ldots$ and let $\left\langle y\right\rangle\in [0,1)$ denote the fractional part of the real number $y$ . Show that there exists a real sequence $\{ x_n\}_{n=1}^{\infty}$ such that the numbers $\langle a_{n,1}x_n \rangle, \ldots, \langle a_{n,n}x_n \rangle$ are asymptotically uniformly distributed on the interval $[0,1]$ .

(translated by L. Erdős)

Let $d>1$ be integer and $0<r<\frac12$ . Show that there exist finitely many (depending only on $d,r$ ) nonzero vectors in $\mathbb{R}^d$ such that if the distance of a straight line in $\mathbb{R}^d$ from the integer lattice $\mathbb{Z}^d$ is at least $r$ , then this line is orthogonal to one of these finitely many vectors.

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Show that the recursion $n=x_n(x_{n-1}+x_n+x_{n+1})$ , $n=1,2,\ldots$ , $x_0=0$ has exaclty one nonnegative solution.

(translated by L. Erdős)

Let $r$ be a nonnegative continuous function on the real line. Show that there exists a function $f\in C^1(\mathbb{R})$ , not identically zero, such that $f'(x)=f(x-r(f(x)))$ , $x\in\mathbb{R}$ .

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Let $f_1, f_2, \ldots$ be continuous real functions on the real line. Is it true that if the series $\sum_{n=1}^{\infty} f_n(x)$ is divergent for every $x$ , then this holds also true for any typical choice of the signs in the sum (i.e. the set of those $\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}$ sequences, for which there series $\sum_{n=1}^{\infty} \epsilon_nf_n(x)$ is convergent at least at one point $x$ , forms a subset of first category within the set $\{+1,-1\}^{\mathbb{N}}$ )?

(translated by L. Erdős)

Given finitely many open half planes on the Euclidean plane. The boundary lines of these half planes divide the plane into convex domains. Find a polynomial $C(q)$ of degree two so that the following holds: for any $q\ge 1$ integer, if the half planes cover each point of the plane at least $q$ times, then the set of points covered exactly $q$ times is the union of at most $C(q)$ domains.

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Let $X$ and $Y$ be independent random variables with "Saint-Petersburg" distribution, i.e. for any $k=1,2,\ldots$ their value is $2^k$ with probability $\frac{1}{2^k}$ . Show that $X$ and $Y$ can be realized on a sufficiently big probability space such that there exists another pair of independent "Saint-Petersburg" random variables $(X', Y')$ on this space with the property that $X+Y=2X'+Y'I(Y'\le X')$ almost surely (here $I(A)$ denotes the indicator function of the event $A$ ).

(translated by L. Erdős)