Navvye/Olympiad-Problems-CSV huggingface dataset:
2007 Baltic Way
For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that

$$\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 $$
2007 Baltic Way
A sequence of integers $a_1,a_2,a_3,\ldots$ is called exact if $a_n^2-a_m^2=a_{n-m}a_{n+m}$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$.
2007 Baltic Way
Suppose that $F,G,H$ are polynomials of degree at most $2n+1$ with real coefficients such that:

i) For all real $x$ we have $F(x)\le G(x)\le H(x)$.

ii) There exist distinct real numbers $x_1,x_2,\ldots ,x_n$ such that $F(x_i)=H(x_i)\quad\text{for}\ i=1,2,3,\ldots ,n$.

iii) There exists a real number $x_0$ different from $x_1,x_2,\ldots ,x_n$ such that $F(x_0)+H(x_0)=2G(x_0)$.

Prove that $F(x)+H(x)=2G(x)$ for all real numbers $x$.
2007 Baltic Way
Let $a_1,a_2,\ldots ,a_n$ be positive real numbers, and let $S=a_1+a_2 +\ldots +a_n$ . Prove that

$$(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2 $$
2007 Baltic Way
A function $f$ is defined on the set of all real numbers except $0$ and takes all real values except $1$. It is also known that



$\color{white}\ . \ \color{black}\ \quad f(xy)=f(x)f(-y)-f(x)+f(y)$



for any $x,y\not= 0$ and that



$\color{white}\ . \ \color{black}\ \quad f(f(x))=\frac{1}{f(\frac{1}{x})}$



for any $x\not\in\{ 0,1\}$. Determine all such functions $f$.
2007 Baltic Way
Freddy writes down numbers $1, 2,\ldots ,n$ in some order. Then he makes a list of all pairs $(i, j)$ such that $1\le i<j\le n$ and the $i$-th number is bigger than the $j$-th number in his permutation. After that, Freddy repeats the following action while possible: choose a pair $(i, j)$ from the current list, interchange the $i$-th and the $j$-th number in the current permutation, and delete $(i, j)$ from the list. Prove that Freddy can choose pairs in such an order that, after the process finishes, the numbers in the permutation are in ascending order.
2007 Baltic Way
A squiggle is composed of six equilateral triangles with side length $1$ as shown in the figure below. Determine all possible integers $n$ such that an equilateral triangle with side length $n$ can be fully covered with squiggles (rotations and reflections of squiggles are allowed, overlappings are not).



[asy]

import graph; size(100); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; 

draw((0,0)--(0.5,1),linewidth(2pt)); draw((0.5,1)--(1,0),linewidth(2pt)); draw((0,0)--(3,0),linewidth(2pt)); draw((1.5,1)--(2,0),linewidth(2pt)); draw((2,0)--(2.5,1),linewidth(2pt)); draw((0.5,1)--(2.5,1),linewidth(2pt)); draw((1,0)--(2,2),linewidth(2pt)); draw((2,2)--(3,0),linewidth(2pt));

dot((0,0),ds); dot((1,0),ds); dot((0.5,1),ds); dot((2,0),ds); dot((1.5,1),ds); dot((3,0),ds); dot((2.5,1),ds); dot((2,2),ds); clip((-4.28,-10.96)--(-4.28,6.28)--(16.2,6.28)--(16.2,-10.96)--cycle);[/asy]
2007 Baltic Way
Call a set $A$ of integers non-isolated, if for every $a\in A$ at least one of the numbers $a-1$ and $a+1$ also belongs to $A$. Prove that the number of five-element non-isolated subsets of $\{1, 2,\ldots ,n\}$ is $(n-4)^2$.
2007 Baltic Way
A society has to elect a board of governors. Each member of the society has chosen $10$ candidates for the board, but he will be happy if at least one of them will be on the board. For each six members of the society there exists a board consisting of two persons making all of these six members happy. Prove that a board consisting of $10$ persons can be elected making every member of the society happy.
2007 Baltic Way
We are given an $18\times 18$ table, all of whose cells may be black or white. Initially all the cells are coloured white. We may perform the following operation: choose one column or one row and change the colour of all cells in this column or row. Is it possible by repeating the operation to obtain a table with exactly $16$ black cells?
2007 Baltic Way
In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements:

i) $P$ is the circumcentre of triangle $ABC$.

ii) All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$.

iii) The oriented segment $PQ$ has the same direction as the oriented segment $AD$. Similarly, $QR$ has the same direction as $BE$, and $Rs$ has the same direction as $CF$.

Prove that $S$ is the incentre of triangle $ABC$.
2007 Baltic Way
Let $M$ be a point on the arc $AB$ of the circumcircle of the triangle $ABC$ which does not contain $C$. Suppose that the projections of $M$ onto the lines $AB$ and $BC$ lie on the sides themselves, not on their extensions. Denote these projections by $X$ and $Y$, respectively. Let $K$ and $N$ be the midpoints of $AC$ and $XY$, respectively. Prove that $\angle MNK=90^{\circ}$ .
2007 Baltic Way
Let $t_1,t_2,\ldots,t_k$ be different straight lines in space, where $k>1$. Prove that points $P_i$ on $t_i$, $i=1,\ldots,k$, exist such that $P_{i+1}$ is the projection of $P_i$ on $t_{i+1}$ for $i=1,\ldots,k-1$, and $P_1$ is the projection of $P_k$ on $t_1$.
2007 Baltic Way
In a convex quadrilateral $ABCD$ we have $ADC = 90^{\circ}$. Let $E$ and $F$ be the projections of $B$ onto the lines $AD$ and $AC$, respectively. Assume that $F$ lies between $A$ and $C$, that $A$ lies between $D$ and $E$, and that the line $EF$ passes through the midpoint of the segment $BD$. Prove that the quadrilateral $ABCD$ is cyclic.
2007 Baltic Way
The incircle of the triangle $ABC$ touches the side $AC$ at the point $D$. Another circle passes through $D$ and touches the rays $BC$ and $BA$, the latter at the point $A$. Determine the ratio $\frac{AD}{DC}$.
2007 Baltic Way
Let $a$ and $b$ be rational numbers such that $s=a+b=a^2+b^2$. Prove that $s$ can be written as a fraction where the denominator is relatively prime to $6$.
2007 Baltic Way
Let $x,y,z$ be positive integers such that $\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$. Prove that $d\le \sqrt[3]{xy+yz+zx}$.
2007 Baltic Way
Let $a,b,c,d$ be non-zero integers, such that the only quadruple of integers $(x, y, z, t)$ satisfying the equation

$$ax^2+by^2+cz^2+dt^2=0$$

is $x=y=z=t=0$. Does it follow that the numbers $a,b,c,d$ have the same sign?
2007 Baltic Way
Let $r$ and $k$ be positive integers such that all prime divisors of $r$ are greater than $50$.



A positive integer, whose decimal representation (without leading zeroes) has at least $k$ digits, will be called nice if every sequence of $k$ consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of $r$.



Prove that if there exist infinitely many nice numbers, then the number $10^k-1$ is nice as well.
2007 Baltic Way
Let $a$ and $b$ be positive integers, $b<a$, such that $a^3+b^3+ab$ is divisible by $ab(a-b)$. Prove that $ab$ is a perfect cube.
2006 Baltic Way
For a sequence $(a_{n})_{n\geq 1}$ of real numbers it is known that $a_{n}=a_{n-1}+a_{n+2}$ for $n\geq 2$.

What is the largest number of its consecutive elements that can all be positive?
2006 Baltic Way
Suppose that the real numbers $a_i\in [-2,17],\ i=1,2,\ldots,59,$ satisfy $a_1+a_2+\ldots+a_{59}=0.$

Prove that

$$a_1^2+a_2^2+\ldots+a_{59}^2\le 2006$$
2006 Baltic Way
Prove that for every polynomial $P(x)$ with real coefficients there exist a positive integer $m$ and polynomials $P_{1}(x),\ldots , P_{m}(x)$ with real coefficients such that

$$P(x) = (P_{1}(x))^{3}+\ldots +(P_{m}(x))^{3}$$
2006 Baltic Way
Let $a,b,c,d,e,f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of



$\color{white}\ .\quad \ \color{black}\ \quad abc+bcd+cde+def+efa+fab$



and determine all $6$-tuples $(a,b,c,d,e,f)$ for which this maximal value is achieved.
2006 Baltic Way
An occasionally unreliable professor has devoted his last book to a certain binary operation $*$. When this operation is applied to any two integers, the result is again an integer. The operation is known to satisfy the following axioms:



$\text{a})\ x*(x*y)=y$ for all $x,y\in\mathbb{Z}$;



$\text{b})\ (x*y)*y=x$ for all $x,y\in\mathbb{Z}$.



The professor claims in his book that



$1.$ The operation $*$ is commutative: $x*y=y*x$ for all $x,y\in\mathbb{Z}$.



$2.$ The operation $*$ is associative: $(x*y)*z=x*(y*z)$ for all $x,y,z\in\mathbb{Z}$.



Which of these claims follow from the stated axioms?
2006 Baltic Way
Determine the maximal size of a set of positive integers with the following properties:



$1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.



$2.$ No digit occurs more than once in the same integer.



$3.$ The digits in each integer are in increasing order.



$4.$ Any two integers have at least one digit in common (possibly at different positions).



$5.$ There is no digit which appears in all the integers.
2006 Baltic Way
A photographer took some pictures at a party with $10$ people. Each of the $45$ possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken?
2006 Baltic Way
The director has found out that six conspiracies have been set up in his department, each of them involving exactly $3$ persons. Prove that the director can split the department in two laboratories so that none of the conspirative groups is entirely in the same laboratory.
2006 Baltic Way
To every vertex of a regular pentagon a real number is assigned. We may perform the following operation repeatedly: we choose two adjacent vertices of the pentagon and replace each of the two numbers assigned to these vertices by their arithmetic mean. Is it always possible to obtain the position in which all five numbers are zeroes, given that in the initial position the sum of all five numbers is equal to zero?
2006 Baltic Way
$162$ pluses and $144$ minuses are placed in a $30\times 30$ table in such a way that each row and each column contains at most $17$ signs. (No cell contains more than one sign.) For every plus we count the number of minuses in its row and for every minus we count the number of pluses in its column. Find the maximum of the sum of these numbers.
2006 Baltic Way
The altitudes of a triangle are $12$, $15$, and $20$.  What is the area of this triangle?
2006 Baltic Way
Let $ABC$ be a triangle, let $B_{1}$ be the midpoint of the side $AB$ and $C_{1}$ the midpoint of the side $AC$. Let $P$ be the point of intersection, other than $A$, of the circumscribed circles around the triangles $ABC_{1}$ and $AB_{1}C$. Let $P_{1}$ be the point of intersection, other than $A$, of the line $AP$ with the circumscribed circle around the triangle $AB_{1}C_{1}$. Prove that $2AP=3AP_{1}$.
2006 Baltic Way
In a triangle $ABC$, points $D,E$ lie on sides $AB,AC$ respectively. The lines $BE$ and $CD$ intersect at $F$. Prove that if



$\color{white}\ .\quad\ \color{black}\ \quad BC^2=BD\cdot BA+CE\cdot CA,$



then the points $A,D,F,E$ lie on a circle.
2006 Baltic Way
There are $2006$ points marked on the surface of a sphere. Prove that the surface can be cut into $2006$ congruent pieces so that each piece contains exactly one of these points inside it.
2006 Baltic Way
Let the medians of the triangle $ABC$ intersect at point $M$. A line $t$ through $M$ intersects the circumcircle of $ABC$ at $X$ and $Y$ so that $A$ and $C$ lie on the same side of $t$. Prove that $BX\cdot BY=AX\cdot AY+CX\cdot CY$.
2006 Baltic Way
Are there $4$ distinct positive integers such that adding the product of any two of them to $2006$ yields a perfect square?
2006 Baltic Way
Determine all positive integers $n$ such that $3^{n}+1$ is divisible by $n^{2}$.
2006 Baltic Way
For a positive integer $n$ let $a_n$ denote the last digit of $n^{(n^n)}$. Prove that the sequence $(a_n)$ is periodic and determine the length of the minimal period.
2006 Baltic Way
Does there exist a sequence $a_1,a_2,a_3,\ldots $ of positive integers such that the sum of every $n$ consecutive elements is divisible by $n^2$ for every positive integer $n$?
2006 Baltic Way
A $12$-digit positive integer consisting only of digits $1,5$ and $9$ is divisible by $37$. Prove that the sum of its digits is not equal to $76$.
2005 Baltic Way
Let $a_0$ be a positive integer. Define the sequence $\{a_n\}_{n \geq 0}$ as follows: if $$ a_n = \sum_{i = 0}^jc_i10^i $$ where $c_i \in \{0,1,2,\cdots,9\}$, then $$ a_{n + 1} = c_0^{2005} + c_1^{2005} + \cdots + c_j^{2005}. $$ Is it possible to choose $a_0$ such that all terms in the sequence are distinct?
2005 Baltic Way
Let $\alpha$, $\beta$ and $\gamma$ be three acute angles such that $\sin \alpha+\sin \beta+\sin \gamma = 1$. Show that

$$\tan^{2}\alpha+\tan^{2}\beta+\tan^{2}\gamma \geq \frac{3}{8}. $$
2005 Baltic Way
Consider the sequence $\{a_k\}_{k \geq 1}$ defined by $a_1 = 1$, $a_2 = \frac{1}{2}$ and $$ a_{k + 2} = a_k + \frac{1}{2}a_{k + 1} + \frac{1}{4a_ka_{k + 1}}\ \textrm{for}\ k \geq 1. $$ Prove that $$ \frac{1}{a_1a_3} + \frac{1}{a_2a_4} + \frac{1}{a_3a_5} + \cdots + \frac{1}{a_{98}a_{100}} < 4. $$
2005 Baltic Way
Find three different polynomials $P(x)$ with real coefficients such that $P\left(x^2 + 1\right) = P(x)^2 + 1$ for all real $x$.
2005 Baltic Way
Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that

$$\frac a{a^{2}+2}+\frac b{b^{2}+2}+\frac c{c^{2}+2}\leq 1 $$
2005 Baltic Way
Let $N$ and $K$ be positive integers satisfying $1 \leq K \leq N$. A deck of $N$ different playing cards is shuffled by repeating the operation of reversing the order of $K$ topmost cards and moving these to the bottom of the deck. Prove that the deck will be back in its initial order after a number of operations not greater than $(2N/K)^2$.
2005 Baltic Way
A rectangular array has $ n$ rows and $ 6$ columns, where $ n \geq 2$. In each cell there is written either $ 0$ or $ 1$. All rows in the array are different from each other. For each two rows $ (x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})$ and $ (y_{1},y_{2},y_{3},y_{4},y_{5},y_{6})$, the row $ (x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{4}y_{4},x_{5}y_{5},x_{6}y_{6})$ can be found in the array as well. Prove that there is a column in which at least half of the entries are zeros.
2005 Baltic Way
Consider a $25 \times 25$ grid of unit squares. Draw with a red pen contours of squares of any size on the grid. What is the minimal number of squares we must draw in order to colour all the lines of the grid?
2005 Baltic Way
A rectangle is divided into $200\times 3$ unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size $1\times 2$ is divisible by $3$.
2005 Baltic Way
Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc=m$.
2005 Baltic Way
Let the points $D$ and $E$ lie on the sides $BC$ and $AC$, respectively, of the triangle $ABC$, satisfying $BD=AE$. The line joining the circumcentres of the triangles $ADC$ and $BEC$ meets the lines $AC$ and $BC$ at $K$ and $L$, respectively. Prove that $KC=LC$.
2005 Baltic Way
Let $ABCD$ be a convex quadrilateral such that $BC=AD$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. The lines $AD$ and $BC$ meet the line $MN$ at $P$ and $Q$, respectively. Prove that $CQ=DP$.
2005 Baltic Way
What the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle

$(a)$ of size $6\times 3$?

$(b)$ of size $5\times 3$?
2005 Baltic Way
Let the medians of the triangle $ABC$ meet at $G$. Let $D$ and $E$ be different points on the line $BC$ such that $DC=CE=AB$, and let $P$ and $Q$ be points on the segments $BD$ and $BE$, respectively, such that $2BP=PD$ and $2BQ=QE$. Determine $\angle PGQ$.
2005 Baltic Way
Let the lines $e$ and $f$ be perpendicular and intersect each other at $H$. Let $A$ and $B$ lie on $e$ and $C$ and $D$ lie on $f$, such that all five points $A,B,C,D$ and $H$ are distinct. Let the lines $b$ and $d$ pass through $B$ and $D$ respectively, perpendicularly to $AC$; let the lines $a$ and $c$ pass through $A$ and $C$ respectively, perpendicularly to $BD$. Let $a$ and $b$ intersect at $X$ and $c$ and $d$ intersect at $Y$. Prove that $XY$ passes through $H$.
2005 Baltic Way
Let $n$ be a positive integer, let $p$ be prime and let $q$ be a divisor of $(n + 1)^p - n^p$. Show that $p$ divides $q - 1$.
2005 Baltic Way
A sequence $(x_n)_{n\ge 0}$ is defined as follows: $x_0=a,x_1=2$ and $x_n=2x_{n-1}x_{n-2}-x_{n-1}-x_{n-2}+1$ for all $n>1$. Find all integers $a$ such that $2x_{3n}-1$ is a perfect square for all $n\ge 1$.
2005 Baltic Way
Let $x$ and $y$ be positive integers and assume that $z=\frac{4xy}{x+y}$ is an odd integer. Prove that at least one divisor of $z$ can be expressed in the form $4n-1$ where $n$ is a positive integer.
2005 Baltic Way
Is it possible to find $2005$ different positive square numbers such that their sum is also a square number ?
2005 Baltic Way
Find all positive integers $n=p_1p_2 \cdots p_k$ which divide $(p_1+1)(p_2+1)\cdots (p_k+1)$ where $p_1 p_2 \cdots p_k$ is the factorization of $n$ into prime factors (not necessarily all distinct).
2004 Baltic Way
Given a sequence $a_1,a_2,\ldots $ of non-negative real numbers satisfying the conditions:



1. $a_n + a_{2n} \geq 3n$;

2. $a_{n+1}+n \leq 2\sqrt{a_n \left(n+1\right)}$



for all $n\in\mathbb N$ (where $\mathbb N=\left\{1,2,3,...\right\}$).



(1) Prove that the inequality $a_n \geq n$ holds for every $n \in \mathbb N$.

(2) Give an example of such a sequence.
2004 Baltic Way
Let $ P(x)$  be a polynomial with a non-negative coefficients. Prove that if the inequality $ P\left(\frac {1}{x}\right)P(x)\geq 1$ holds for $ x = 1$, then this inequality holds for each positive $ x$.
2004 Baltic Way
Let $p, q, r$ be positive real numbers and $n$ a natural number. Show that if $pqr = 1$, then $$ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+1} \leq 1. $$
2004 Baltic Way
Let $x_1$, $x_2$, ..., $x_n$ be real numbers with arithmetic mean $X$. Prove that there is a positive integer $K$ such that for any integer $i$ satisfying $0\leq i<K$, we have $\frac{1}{K-i}\sum_{j=i+1}^{K} x_j \leq X$. (In other words, prove that there is a positive integer $K$ such that the arithmetic mean of each of the lists $\left\{x_1,x_2,...,x_K\right\}$, $\left\{x_2,x_3,...,x_K\right\}$, $\left\{x_3,...,x_K\right\}$, ..., $\left\{x_{K-1},x_K\right\}$, $\left\{x_K\right\}$ is not greater than $X$.)
2004 Baltic Way
Determine the range of the following function defined for  integer $k$,



$$f(k)=(k)_3+(2k)_5+(3k)_7-6k$$



where $(k)_{2n+1}$ denotes the multiple of $2n+1$ closest to $k$
2004 Baltic Way
A positive integer is written on each of the six faces of a cube. For each vertex of the cube we compute the product of the numbers on the three adjacent faces. The sum of these products is $1001$. What is the sum of the six numbers on the faces?
2004 Baltic Way
Find all sets $X$ consisting of at least two positive integers such that for every two elements $m,n\in X$, where $n>m$, there exists an element $k\in X$ such that $n=mk^2$.
2004 Baltic Way
Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.
2004 Baltic Way
A set $S$ of $n-1$ natural numbers is given ($n\ge 3$). There exist at least at least two elements in this set whose difference is not divisible by $n$. Prove that it is possible to choose a non-empty subset of $S$ so that the sum of its elements is divisible by $n$.
2004 Baltic Way
Is there an infinite sequence of prime numbers $p_1$, $p_2$, $\ldots$, $p_n$, $p_{n+1}$, $\ldots$ such that $|p_{n+1}-2p_n|=1$ for each $n \in \mathbb{N}$?
2004 Baltic Way
Given a table $m\times n$, in each cell of which a number $+1$ or $-1$ is written. It is known that initially exactly one $-1$ is in the table, all the other numbers being $+1$. During a move, it is allowed to chose any cell containing $-1$, replace this $-1$ by $0$, and simultaneously multiply all the numbers in the neighbouring cells by $-1$ (we say that two cells are neighbouring if they have a common side). Find all $(m,n)$ for which using such moves one can obtain the table containing zeros only, regardless of the cell in which the initial $-1$ stands.
2004 Baltic Way
There are $2n$ different numbers in a row. By one move we can interchange any two numbers or interchange any $3$ numbers cyclically (choose $a,b,c$ and place $a$ instead of $b$, $b$ instead of $c$, $c$ instead of $a$). What is the minimal number of moves that is always sufficient to arrange the numbers in increasing order ?
2004 Baltic Way
The $25$ member states of the European Union set up a committee with the following rules:

1) the committee should meet daily;

2) at each meeting, at least one member should be represented;

3) at any two different meetings, a different set of member states should be represented;

4) at $n^{th}$ meeting, for every $k<n$, the set of states represented should include at least one state that was represented at the $k^{th}$ meeting.



For how many days can the committee have its meetings?
2004 Baltic Way
We say that a pile is a set of four or more nuts. Two persons play the following game. They start with one pile of $n \geq 4$ nuts. During a move a player takes one of the piles that they have and split it into two nonempty sets (these sets are not necessarily piles, they can contain arbitrary number of nuts). If the player cannot move, he loses. For which values of $n$ does the first player have a winning strategy?
2004 Baltic Way
A circle is divided into $13$ segments, numbered consecutively from $1$ to $13$. Five fleas called $A,B,C,D$ and $E$ are sitting in the segments $1,2,3,4$ and $5$. A flea is allowed to jump to an empty segment five positions away in either direction around the circle. Only one flea jumps at the same time, and two fleas cannot be in the same segment. After some jumps, the fleas are back in the segments $1,2,3,4,5$, but possibly in some other order than they started. Which orders are possible ?
2004 Baltic Way
Through a point $P$ exterior to a given circle pass a secant and a tangent to the circle. The secant intersects the circle at $A$ and $B$, and the tangent touches the circle at $C$ on the same side of the diameter through $P$ as the points $A$ and $B$. The projection of the point $C$ on the diameter is called $Q$. Prove that the line $QC$ bisects the angle $\angle AQB$.
2004 Baltic Way
Consider a rectangle with sidelengths 3 and 4, pick an arbitrary inner point on each side of this rectangle. Let $x, y, z$ and $u$ denote the side lengths of the quadrilateral spanned by these four points. Prove that $25 \leq x^2+y^2+z^2+u^2 \leq 50$.
2004 Baltic Way
A ray emanating from the vertex $A$ of the triangle $ABC$ intersects the side $BC$ at $X$ and the circumcircle of triangle $ABC$ at $Y$. Prove that $\frac{1}{AX}+\frac{1}{XY}\geq \frac{4}{BC}$.
2004 Baltic Way
Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$. Let $M$ be a point on the side $BC$ such that $\angle BAM = \angle DAC$. Further, let $L$ be the second intersection point of the circumcircle of the triangle $CAM$ with the side $AB$, and let $K$ be the second intersection point of the circumcircle of the triangle $BAM$ with the side $AC$. Prove that $KL \parallel BC$.
2004 Baltic Way
Three fixed circles pass through the points $A$ and $B$. Let $X$ be a variable point on the first circle different from $A$ and $B$. The line $AX$ intersects the other two circles at $Y$ and $Z$ (with $Y$ between $X$ and $Z$). Show that the ratio $\frac{XY}{YZ}$ is independent of the position of $X$.
2003 Baltic Way
Find all functions $f:\mathbb{Q}^{+}\rightarrow \mathbb{Q}^{+}$ which for all $x \in \mathbb{Q}^{+}$ fulfil

$$f\left(\frac{1}{x}\right)=f(x) \ \ \text{and} \ \ \left(1+\frac{1}{x}\right)f(x)=f(x+1). $$