BALTIC-WAY 2003

Let $\mathbb{Q}_{+}$be the set of positive rational numbers. Find all functions $f: \mathbb{Q}_{+} \rightarrow \mathbb{Q}_{+}$which for all $x \in \mathbb{Q}_{+}$fulfil

(1) $f\left(\frac{1}{x}\right)=f(x)$

(2) $\left(1+\frac{1}{x}\right) f(x)=f(x+1)$

Prove that any real solution of

$$x^{3}+p x+q=0$$

satisfies the inequality $4 q x \leq p^{2}$.

Let $x, y$ and $z$ be positive real numbers such that $x y z=1$. Prove that

$$(1+x)(1+y)(1+z) \geq 2\left(1+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}+\sqrt[3]{\frac{x}{z}}\right)$$

Let $a, b, c$ be positive real numbers. Prove that

$$\frac{2 a}{a^{2}+b c}+\frac{2 b}{b^{2}+c a}+\frac{2 c}{c^{2}+a b} \leq \frac{a}{b c}+\frac{b}{c a}+\frac{c}{a b} .$$

A sequence $\left(a_{n}\right)$ is defined as follows: $a_{1}=\sqrt{2}, a_{2}=2$, and $a_{n+1}=a_{n} a_{n-1}^{2}$ for $n \geq 2$. Prove that for every $n \geq 1$ we have

$$\left(1+a_{1}\right)\left(1+a_{2}\right) \cdots\left(1+a_{n}\right)<(2+\sqrt{2}) a_{1} a_{2} \cdots a_{n} .$$

Let $n \geq 2$ and $d \geq 1$ be integers with $d \mid n$, and let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers such that $x_{1}+x_{2}+\cdots+x_{n}=0$. Prove that there are at least $\left(\begin{array}{l}n-1 \\ d-1\end{array}\right)$ choices of $d$ indices $1 \leq i_{1}<i_{2}<\cdots<i_{d} \leq n$ such that $x_{i_{1}}+x_{i_{2}}+\cdots+x_{i_{d}} \geq 0$.

Let $X$ be a subset of $\{1,2,3, \ldots, 10000\}$ with the following property: If $a, b \in X, a \neq b$, then $a \cdot b \notin X$. What is the maximal number of elements in $X$ ?

There are 2003 pieces of candy on a table. Two players alternately make moves. A move consists of eating one candy or half of the candies on the table (the "lesser half" if there is an odd number of candies); at least one candy must be eaten at each move. The loser is the one who eats the last candy. Which player - the first or the second - has a winning strategy?

It is known that $n$ is a positive integer, $n \leq 144$. Ten questions of type "Is $n$ smaller than a?" are allowed. Answers are given with a delay: The answer to the $i$ 'th question is given only after the $(i+1)$ 'st question is asked, $i=1,2, \ldots, 9$. The answer to the tenth question is given immediately after it is asked. Find a strategy for identifying $n$.

A lattice point in the plane is a point whose coordinates are both integral. The centroid of four points $\left(x_{i}, y_{i}\right), i=1,2,3,4$, is the point $\left(\frac{x_{1}+x_{2}+x_{3}+x_{4}}{4}, \frac{y_{1}+y_{2}+y_{3}+y_{4}}{4}\right)$. Let $n$ be the largest natural number with the following property: There are $n$ distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that $n=12$.

Is it possible to select 1000 points in a plane so that at least 6000 distances between two of them are equal?

Let $A B C D$ be a square. Let $M$ be an inner point on side $B C$ and $N$ be an inner point on side $C D$ with $\angle M A N=45^{\circ}$. Prove that the circumcentre of $A M N$ lies on $A C$.

Let $A B C D$ be a rectangle and $B C=2 \cdot A B$. Let $E$ be the midpoint of $B C$ and $P$ an arbitrary inner point of $A D$. Let $F$ and $G$ be the feet of perpendiculars drawn correspondingly from $A$ to $B P$ and from $D$ to $C P$. Prove that the points $E, F, P, G$ are concyclic.

Let $A B C$ be an arbitrary triangle and $A M B, B N C, C K A$ regular triangles outward of $A B C$. Through the midpoint of $M N$ a perpendicular to $A C$ is constructed; similarly through the midpoints of $N K$ resp. KM perpendiculars to $A B$ resp. $B C$ are constructed. Prove that these three perpendiculars intersect at the same point.

Let $P$ be the intersection point of the diagonals $A C$ and $B D$ in a cyclic quadrilateral. $A$ circle through $P$ touches the side $C D$ in the midpoint $M$ of this side and intersects the segments $B D$ and $A C$ in the points $Q$ and $R$, respectively. Let $S$ be a point on the segment $B D$ such that $B S=D Q$. The parallel to $A B$ through $S$ intersects $A C$ at $T$. Prove that $A T=R C$.

Find all pairs of positive integers $(a, b)$ such that $a-b$ is a prime and $a b$ is a perfect square.

All the positive divisors of a positive integer $n$ are stored into an array in increasing order. Mary has to write a program which decides for an arbitrarily chosen divisor $d>1$ whether it is a prime. Let $n$ have $k$ divisors not greater than $d$. Mary claims that it suffices to check divisibility of $d$ by the first $\lceil k / 2\rceil$ divisors of $n$ : If a divisor of $d$ greater than 1 is found among them, then $d$ is composite, otherwise d is prime. Is Mary right?

Every integer is coloured with exactly one of the colours BLUE, GREEN, RED, YELLOW. Can this be done in such a way that if $a, b, c, d$ are not all 0 and have the same colour, then $3 a-2 b \neq 2 c-3 d$ ?

Let $a$ and $b$ be positive integers. Prove that if $a^{3}+b^{3}$ is the square of an integer, then $a+b$ is not a product of two different prime numbers.

Let $n$ be a positive integer such that the sum of all the positive divisors of $n$ (except $n$ ) plus the number of these divisors is equal to $n$. Prove that $n=2 m^{2}$ for some integer $m$.