BALTIC-WAY 2011

The real numbers $x_1,\ldots ,x_{2011}$ satisfy $$x_1+x_2=2x_1',\ x_2+x_3=2x_2', \ \ldots, \ x_{2011}+x_1=2x_{2011}'$$ where $x_1',x_2',\ldots,x_{2011}'$ is a permutation of $x_1,x_2,\ldots,x_{2011}$ . Prove that $x_1=x_2=\ldots =x_{2011}$ .

Let $f:\mathbb{Z}\to\mathbb{Z}$ be a function such that for all integers $x$ and $y$ , the following holds: $$f(f(x)-y)=f(y)-f(f(x)).$$ Show that $f$ is bounded.

A sequence $a_1,a_2,a_3,\ldots$ of non-negative integers is such that $a_{n+1}$ is the last digit of $a_n^n+a_{n-1}$ for all $n>2$ . Is it always true that for some $n_0$ the sequence $a_{n_0},a_{n_0+1},a_{n_0+2},\ldots$ is periodic?

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $$f(f(x))=x^2-x+1$$ for all real numbers $x$ . Determine $f(0)$ .

Let $n$ be a positive integer. Prove that the number of lines which go through the origin and precisely one other point with integer coordinates $(x,y),0\le x,y\le n$ , is at least $\frac{n^2}{4}$ .

Let $T$ denote the $15$ -element set $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$ . Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$ .

In Greifswald there are three schools called $A,B$ and $C$ , each of which is attended by at least one student. Among any three students, one from $A$ , one from $B$ and one from $C$ , there are two knowing each other and two not knowing each other. Prove that at least one of the following holds:

• Some student from $A$ knows all students from $B$ .
• Some student from $B$ knows all students from $C$ .
• Some student from $C$ knows all students from $A$ .

Given a rectangular grid, split into $m\times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions:

• All squares touching the border of the grid are coloured black.
• No four squares forming a $2\times 2$ square are coloured in the same colour.
• No four squares forming a $2\times 2$ square are coloured in such a way that only diagonally touching squares have the same colour.

Which grid sizes $m\times n$ (with $m,n\ge 3$ ) have a valid colouring?

Two persons play the following game with integers. The initial number is $2011^{2011}$ . The players move in turns. Each move consists of subtraction of an integer between $1$ and $2010$ inclusive, or division by $2011$ , rounding down to the closest integer when necessary. The player who first obtains a non-positive integer wins. Which player has a winning strategy?

Let $AB$ and $CD$ be two diameters of the circle $C$ . For an arbitrary point $P$ on $C$ , let $R$ and $S$ be the feet of the perpendiculars from $P$ to $AB$ and $CD$ , respectively. Show that the length of $RS$ is independent of the choice of $P$ .

Let $P$ be a point inside a square $ABCD$ such that $PA:PB:PC$ is $1:2:3$ . Determine the angle $\angle BPA$ .

Let $E$ be an interior point of the convex quadrilateral $ABCD$ . Construct triangles $\triangle ABF,\triangle BCG,\triangle CDH$ and $\triangle DAI$ on the outside of the quadrilateral such that the similarities $\triangle ABF\sim\triangle DCE,\triangle BCG\sim \triangle ADE,\triangle CDH\sim\triangle BAE$ and $\triangle DAI\sim\triangle CBE$ hold. Let $P,Q,R$ and $S$ be the projections of $E$ on the lines $AB,BC,CD$ and $DA$ , respectively. Prove that if the quadrilateral $PQRS$ is cyclic, then $$EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.$$

The incircle of a triangle $ABC$ touches the sides $BC,CA,AB$ at $D,E,F$ , respectively. Let $G$ be a point on the incircle such that $FG$ is a diameter. The lines $EG$ and $FD$ intersect at $H$ . Prove that $CH\parallel AB$ .

Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$ . Suppose that a point $E$ on the side $AD$ satisfies the equality $$AE\cdot ED + BE^2=CD\cdot AE.$$ Show that $\angle EBA=\angle DCB$ .

Let $a$ be any integer. Define the sequence $x_0,x_1,\ldots$ by $x_0=a$ , $x_1=3$ , and for all $n>1$ $$x_n=2x_{n-1}-4x_{n-2}+3.$$ Determine the largest integer $k_a$ for which there exists a prime $p$ such that $p^{k_a}$ divides $x_{2011}-1$ .

Determine all positive integers $d$ such that whenever $d$ divides a positive integer $n$ , $d$ will also divide any integer obtained by rearranging the digits of $n$ .

Determine all pairs $(p,q)$ of primes for which both $p^2+q^3$ and $q^2+p^3$ are perfect squares.

Let $p\neq 3$ be a prime number. Show that there is a non-constant arithmetic sequence of positive integers $x_1,x_2,\ldots ,x_p$ such that the product of the terms of the sequence is a cube.

An integer $n\ge 1$ is called balanced if it has an even number of distinct prime divisors. Prove that there exist infinitely many positive integers $n$ such that there are exactly two balanced numbers among $n,n+1,n+2$ and $n+3$ .