Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 Baltic Way
Two players $A$ and $B$ play the following game.  Before the game starts, $A$ chooses 1000 not necessarily different odd primes, and then $B$ chooses half of them and writes them on a blackboard.  In each turn a player chooses a positive integer $n$, erases some primes $p_1$, $p_2$, $\dots$, $p_n$ from the blackboard and writes all the prime factors of $p_1 p_2 \dotsm p_n - 2$ instead (if a prime occurs several times in the prime factorization of $p_1 p_2 \dotsm p_n - 2$, it is written as many times as it occurs).  Player $A$ starts, and the player whose move leaves the blackboard empty loses the game.  Prove that one of the two players has a winning strategy and determine who.



Remark: Since 1 has no prime factors, erasing a single 3 is a legal move.
2012 Baltic Way
Let $ABC$ be a triangle with $\angle A = 60^\circ$.  The point $T$ lies inside the triangle in such a way that $\angle ATB = \angle BTC = \angle CTA = 120^\circ$.  Let $M$ be the midpoint of $BC$.  Prove that $TA + TB + TC = 2AM$.
2012 Baltic Way
Let $P_0$, $P_1$, $\dots$, $P_8 = P_0$ be successive points on a circle and $Q$ be a point inside the polygon $P_0 P_1 \dotsb P_7$ such that $\angle P_{i - 1} QP_i = 45^\circ$ for $i = 1$, $\dots$, 8.  Prove that the sum

$$\sum_{i = 1}^8 P_{i - 1} P_i^2$$

is minimal if and only if $Q$ is the centre of the circle.
2012 Baltic Way
Let $ABC$ be an acute triangle, and let $H$ be its orthocentre.  Denote by $H_A$, $H_B$, and $H_C$ the second intersection of the circumcircle with the altitudes from $A$, $B$, and $C$ respectively.  Prove that the area of triangle $H_A H_B H_C$ does not exceed the area of triangle $ABC$.
2012 Baltic Way
Given a triangle $ABC$, let its incircle touch the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively.  Let $G$ be the midpoint of the segment $DE$.  Prove that $\angle EFC = \angle GFD$.
2012 Baltic Way
The circumcentre $O$ of a given cyclic quadrilateral $ABCD$ lies inside the quadrilateral but not on the diagonal $AC$.  The diagonals of the quadrilateral intersect at $I$.  The circumcircle of the triangle $AOI$ meets the sides $AD$ and $AB$ at points $P$ and $Q$, respectively; the circumcircle of the triangle $COI$ meets the sides $CB$ and $CD$ at points $R$ and $S$, respectively.  Prove that $PQRS$ is a parallelogram.
2012 Baltic Way
Let $n$, $m$, and $k$ be positive integers satisfying $(n - 1)n(n + 1) = m^k$.  Prove that $k = 1$.
2012 Baltic Way
Let $d(n)$ denote the number of positive divisors of $n$.  Find all triples $(n,k,p)$, where $n$ and $k$ are positive integers and $p$ is a prime number, such that

$$n^{d(n)} - 1 = p^k.$$
2012 Baltic Way
Find all triples $(a,b,c)$ of integers satisfying $a^2 + b^2 + c^2 = 20122012$.
2012 Baltic Way
Show that $n^n + (n + 1)^{n + 1}$ is composite for infinitely many positive integers $n$.
2012 Baltic Way
Find all integer solutions of the equation $2x^6 + y^7 = 11$.
2011 Baltic Way
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 $x_1=x_2=\ldots =x_{2011}$ .
2011 Baltic Way
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.
2011 Baltic Way
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?
2011 Baltic Way
Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=4$. Prove the inequality

$$\frac{a}{a^3+8}+\frac{b}{b^3+8}+\frac{c}{c^3+8}+\frac{d}{d^3+8}\le\frac{4}{9}$$
2011 Baltic Way
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 $f(0)$.
2011 Baltic Way
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 $\frac{n^2}{4}$.
2011 Baltic Way
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 $S$.
2011 Baltic Way
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 $A$.
2011 Baltic Way
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?
2011 Baltic Way
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?
2011 Baltic Way
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 $P$.
2011 Baltic Way
Let $P$ be a point inside a square $ABCD$ such that $PA:PB:PC$ is $1:2:3$. Determine the angle $\angle BPA$.
2011 Baltic Way
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

$$EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.$$
2011 Baltic Way
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 $CH\parallel AB$.
2011 Baltic Way
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 $\angle EBA=\angle DCB$.
2011 Baltic Way
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 $x_{2011}-1$.
2011 Baltic Way
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 $n$.
2011 Baltic Way
Determine all pairs $(p,q)$ of primes for which both $p^2+q^3$ and $q^2+p^3$ are perfect squares.
2011 Baltic Way
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.
2011 Baltic Way
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 $n+3$.
2010 Baltic Way
Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations

$$\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}$$
2010 Baltic Way
Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that

$$\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1$$
2010 Baltic Way
Let $x_1, x_2, \ldots ,x_n(n\ge 2)$ be real numbers greater than $1$. Suppose that $|x_i-x_{i+1}|<1$ for $i=1, 2,\ldots ,n-1$. Prove that

$$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots +\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}<2n-1$$
2010 Baltic Way
Find all polynomials $P(x)$ with real coefficients such that

$$(x-2010)P(x+67)=xP(x) $$

for every integer $x$.
2010 Baltic Way
Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that

$$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$$

for all $x,y\in\mathbb{R}$.
2010 Baltic Way
An $n\times n$ board is coloured in $n$ colours such that the main diagonal (from top-left to bottom-right) is coloured in the first colour; the two adjacent diagonals are coloured in the second colour; the two next diagonals (one from above and one from below) are coloured in the third colour, etc; the two corners (top-right and bottom-left) are coloured in the $n$-th colour. It happens that it is possible to place on the board $n$ rooks, no two attacking each other and such that no two rooks stand on cells of the same colour. Prove that $n=0\pmod{4}$ or $n=1\pmod{4}$.
2010 Baltic Way
There are some cities in a country; one of them is the capital. For any two cities $A$ and $B$ there is a direct flight from $A$ to $B$ and a direct flight from $B$ to $A$, both having the same price. Suppose that all round trips with exactly one landing in every city have the same total cost. Prove that all round trips that miss the capital and with exactly one landing in every remaining city cost the same.
2010 Baltic Way
In a club with $30$ members, every member initially had a hat. One day each member sent his hat to a different member (a member could have received more than one hat). Prove that there exists a group of $10$ members such that no one in the group has received a hat from another one in the group.
2010 Baltic Way
There is a pile of $1000$ matches. Two players each take turns and can take $1$ to $5$ matches. It is also allowed at most $10$ times during the whole game to take $6$ matches, for example $7$ exceptional moves can be done by the first player and $3$ moves by the second and then no more exceptional moves are allowed. Whoever takes the last match wins. Determine which player has a winning strategy.
2010 Baltic Way
Let $n$ be an integer with $n\ge 3$. Consider all dissections of a convex $n$-gon into triangles by $n-3$ non-intersecting diagonals, and all colourings of the triangles with black and white so that triangles with a common side are always of a different colour. Find the least possible number of black triangles.
2010 Baltic Way
Let $ABCD$ be a square and let $S$ be the point of intersection of its diagonals $AC$ and $BD$. Two circles $k,k'$ go through $A,C$ and $B,D$; respectively. Furthermore, $k$ and $k'$ intersect in exactly two different points $P$ and $Q$. Prove that $S$ lies on $PQ$.
2010 Baltic Way
Let $ABCD$ be a convex quadrilateral with precisely one pair of parallel sides.

$(a)$ Show that the lengths of its sides $AB,BC,CD, DA$ (in this order) do not form an arithmetic progression.

$(b)$ Show that there is such a quadrilateral for which the lengths of its sides $AB ,BC,CD,DA$ form an arithmetic progression after the order of the lengths is changed.
2010 Baltic Way
In an acute triangle $ABC$, the segment $CD$ is an altitude and $H$ is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle $DHB$, determine all possible values of $\angle CAB$.
2010 Baltic Way
Assume that all angles of a triangle $ABC$ are acute. Let $D$ and $E$ be points on the sides $AC$ and $BC$ of the triangle such that $A, B, D,$ and $E$ lie on the same circle. Further suppose the circle through $D,E,$ and $C$ intersects the side $AB$ in two points $X$ and $Y$. Show that the midpoint of $XY$ is the foot of the altitude from $C$ to $AB$.
2010 Baltic Way
The points $M$ and $N$ are chosen on the angle bisector $AL$ of a triangle $ABC$ such that $\angle ABM=\angle ACN=23^{\circ}$. $X$ is a point inside the triangle such that $BX=CX$ and $\angle BXC=2\angle BML$. Find $\angle MXN$.
2010 Baltic Way
For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be amusing if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than $1$?
2010 Baltic Way
Find all positive integers $n$ such that the decimal representation of $n^2$ consists of odd digits only.
2010 Baltic Way
Let $p$ be a prime number. For each $k$, $1\le k\le p-1$, there exists a unique integer denoted by $k^{-1}$ such that $1\le k^{-1}\le p-1$ and $k^{-1}\cdot k=1\pmod{p}$. Prove that the sequence

$$1^{-1},\quad 1^{-1}+2^{-1},\quad 1^{-1}+2^{-1}+3^{-1},\quad \ldots ,\quad 1^{-1}+2^{-1}+\ldots +(p-1)^{-1} $$

(addition modulo $p$) contains at most $\frac{p+1}{2}$ distinct elements.
2010 Baltic Way
For which $k$ do there exist $k$ pairwise distinct primes $p_1,p_2,\ldots ,p_k$ such that

$$p_1^2+p_2^2+\ldots +p_k^2=2010? $$
2010 Baltic Way
Determine all positive integers $n$ for which there exists an infinite subset $A$ of the set $\mathbb{N}$ of positive integers such that for all pairwise distinct $a_1,\ldots , a_n \in A$ the numbers $a_1+\ldots +a_n$ and $a_1a_2\ldots a_n$ are coprime.
2009 Baltic Way
A polynomial $p(x)$ of degree $n\ge 2$ has exactly $n$ real roots, counted with multiplicity. We know that the coefficient of $x^n$ is $1$, all the roots are less than or equal to $1$, and $p(2)=3^n$. What values can $p(1)$ take?
2009 Baltic Way
Let $ a_1,a_{2},\ldots ,a_{100}$ be nonnegative integers satisfying the inequality

$$a_1\cdot (a_1-1)\cdot\ldots\cdot (a_1-20)+a_2\cdot (a_2-1)\cdot\ldots\cdot (a_2-20)+\\ \ldots+a_{100}\cdot (a_{100}-1)\cdot\ldots\cdot (a_{100}-20)\le 100\cdot 99\cdot 98\cdot\ldots\cdot 79.$$

Prove that $a_1+a_2+\ldots+a_{100}\le 9900$.
2009 Baltic Way
Let $ n$ be a given positive integer. Show that we can choose numbers $ c_k\in\{-1,1\}$ ($ i\le k\le n$) such that $$ 0\le\sum_{k=1}^nc_k\cdot k^2\le4.$$
2009 Baltic Way
Determine all integers $ n>1$ for which the inequality $$ x_1^2+x_2^2+\ldots+x_n^2\ge(x_1+x_2+\ldots+x_{n-1})x_n$$ holds for all real $ x_1,x_2,\ldots,x_n$.
2009 Baltic Way
Let $f_0=f_1=1$ and $f_{i+2}=f_{i+1}+f_i$ for all $n\ge 0$. Find all real solutions to the equation

$$x^{2010}=f_{2009}\cdot x+f_{2008}$$
2009 Baltic Way
Let $ a$ and $ b$ be integers such that the equation $ x^3-ax^2-b=0$ has three integer roots. Prove that $ b=dk^2$, where $ d$ and $ k$ are integers and $ d$ divides $ a$.
2009 Baltic Way
Suppose that for a prime number $p$ and integers $a,b,c$ the following holds:

$$6\mid p+1,\quad p\mid a+b+c,\quad p\mid a^4+b^4+c^4.$$

Prove that $p\mid a,b,c$.
2009 Baltic Way
Determine all positive integers $n$ for which there exists a partition of the set

$$\{n,n+1,n+2,\ldots ,n+8\}$$

into two subsets such that the product of all elements of the first subset is equal to the product of all elements of the second subset.
2009 Baltic Way
Determine all positive integers $n$ for which $2^{n+1}-n^2$ is a prime number.
2009 Baltic Way
Let $d(k)$ denote the number of positive divisors of a positive integer $k$. Prove that there exist infinitely many positive integers $M$ that cannot be written as

$$M=\left(\frac{2\sqrt{n}}{d(n)}\right)^2$$

for any positive integer $n$.
2009 Baltic Way
Let $M$ be the midpoint of the side $AC$ of a triangle $ABC$, and let $K$ be a point on the ray $BA$ beyond $A$. The line $KM$ intersects the side $BC$ at the point $L$. $P$is the point on the segment $BM$ such that $PM$ is the bisector of the angle $LPK$. The line $\ell$ passes through $A$ and is parallel to $BM$. Prove that the projection of the point $M$ onto the line $\ell$ belongs to the line $PK$.
2009 Baltic Way
In a quadrilateral $ABCD$ we have $AB||CD$ and $AB=2CD$. A line $\ell$ is perpendicular to $CD$ and contains the point $C$. The circle with centre $D$ and radius $DA$ intersects the line $\ell$ at points $P$ and $Q$. Prove that $AP\perp BQ$.
2009 Baltic Way
The point $H$ is the orthocentre of a triangle $ABC$, and the segments $AD,BE,CF$ are its altitudes. The points $I_1,I_2,I_3$ are the incentres of the triangles $EHF,FHD,DHE$ respectively. Prove that the lines $AI_1,BI_2,CI_3$ intersect at a single point.
2009 Baltic Way
For which $n\ge 2$ is it possible to find $n$ pairwise non-similar triangles $A_1, A_2,\ldots , A_n$ such that each of them can be divided into $n$ pairwise non-similar triangles, each of them similar to one of $A_1,A_2 ,\ldots ,A_n$?
2009 Baltic Way
A unit square is cut into $m$ quadrilaterals $Q_1,\ldots ,Q_m$. For each $i=1,\ldots ,m$ let $S_i$ be the sum of the squares of the four sides of $Q_i$. Prove that

$$S_1+\ldots +S_m\ge 4$$
2009 Baltic Way
A $n$-trønder walk is a walk starting at $(0, 0)$, ending at $(2n, 0)$ with no self intersection and not leaving the first quadrant, where every step is one of the vectors $(1, 1)$, $(1, -1)$ or $(-1, 1)$. Find the number of $n$-trønder walks.
2009 Baltic Way
Find the largest integer $n$ for which there exist $n$ different integers such that none of them are divisible by either of $7,11$ or $13$, but the sum of any two of them is divisible by at least one of $7,11$ and $13$.
2009 Baltic Way
Let $n>2$ be an integer. In a country there are $n$ cities and every two of them are connected by a direct road. Each road is assigned an integer from the set $\{1, 2,\ldots ,m\}$ (different roads may be assigned the same number). The priority of a city is the sum of the numbers assigned to roads which lead to it. Find the smallest $m$ for which it is possible that all cities have a different priority.
2009 Baltic Way
In a party of eight people, each pair of people either know each other or do not know each other. Each person knows exactly three of the others. Determine whether the following two conditions can be satisfied simultaneously:



– for any three people, at least two do not know each other;

– for any four people there are at least two who know each other.
2009 Baltic Way
In the future city Baltic Way there are sixteen hospitals. Every night exactly four of them must be on duty for emergencies. Is it possible to arrange the schedule in such a way that after twenty nights every pair of hospitals have been on common duty exactly once?
2008 Baltic Way
Determine all polynomials $p(x)$ with real coefficients such that $p((x+1)^3)=(p(x)+1)^3$ and $p(0)=0$.
2008 Baltic Way
Prove that if the real numbers $a,b$ and $c$ satisfy $a^2+b^2+c^2=3$ then

$$\frac{a^2}{2+b+c^2}+\frac{b^2}{2+c+a^2}+\frac{c^2}{2+a+b^2}\ge\frac{(a+b+c)^2}{12}$$

When does the inequality hold?
2008 Baltic Way
Does there exist an angle $ \alpha\in(0,\pi/2)$ such that $ \sin\alpha$, $ \cos\alpha$, $ \tan\alpha$ and $ \cot\alpha$, taken in some order, are consecutive terms of an arithmetic progression?
2008 Baltic Way
The polynomial $P$ has integer coefficients and $P(x)=5$ for five different integers $x$. Show that there is no integer $x$ such that $-6\le P(x)\le 4$ or $6\le P(x)\le 16$.
2008 Baltic Way
Suppose that Romeo and Juliet each have a regular tetrahedron to the vertices of which some positive real numbers are assigned. They associate each edge of their tetrahedra with the product of the two numbers assigned to its end points. Then they write on each face of their tetrahedra the sum of the three numbers associated to its three edges. The four numbers written on the faces of Romeo's tetrahedron turn out to coincide with the four numbers written on Juliet's tetrahedron. Does it follow that the four numbers assigned to the vertices of Romeo's tetrahedron are identical to the four numbers assigned to the vertices of Juliet's tetrahedron?
2008 Baltic Way
Find all finite sets of positive integers with at least two elements such that for any two numbers $ a$, $ b$ ($ a > b$) belonging to the set, the number $ \frac {b^2}{a - b}$ belongs to the set, too.
2008 Baltic Way
How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} = \frac 1m + \frac 1n$?
2008 Baltic Way
Consider a set $ A$ of positive integers such that the least element of $ A$ equals $ 1001$ and the product of all elements of $ A$ is a perfect square. What is the least possible value of the greatest element of $ A$?
2008 Baltic Way
Suppose that the positive integers $ a$ and $ b$ satisfy the equation $ a^b-b^a=1008$ Prove that $ a$ and $ b$ are congruent modulo 1008.
2008 Baltic Way
For a positive integer $ n$, let $ S(n)$ denote the sum of its digits. Find the largest possible value of the expression $ \frac {S(n)}{S(16n)}$.
2008 Baltic Way
Consider a subset $A$ of $84$ elements of the set $\{1,\,2,\,\dots,\,169\}$ such that no two elements in the set add up to $169$. Show that $A$ contains a perfect square.
2008 Baltic Way
In  a school class with $ 3n$ children, any two children make a common present to exactly one other child. Prove that for all odd $ n$ it is possible that the following holds: For any three children $ A$, $ B$ and $ C$ in the class, if $ A$ and $ B$ make a present to $ C$ then $ A$ and $ C$ make a present to $ B$.
2008 Baltic Way
For an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened:

i) Every country voted for exactly three problems.

ii) Any two countries voted for different sets of problems.

iii) Given any three countries, there was a problem none of them voted for.

Find the maximal possible number of participating countries.
2008 Baltic Way
Is it possible to build a $ 4\times 4\times4$ cube from blocks of the following shape consisting of $ 4$ unit cubes?
2008 Baltic Way
Some $1\times 2$ dominoes, each covering two adjacent unit squares, are placed on a board of size $n\times n$ such that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is $2008$, find the least possible value of $n$.
2008 Baltic Way
Let $ABCD$ be a parallelogram. The circle with diameter $AC$ intersects the line $BD$ at points $P$ and $Q$. The perpendicular to the line $AC$ passing through the point $C$ intersects the lines $AB$ and $AD$ at points $X$ and $Y$, respectively. Prove that the points $P,Q,X$ and $Y$ lie on the same circle.
2008 Baltic Way
Assume that $ a$, $ b$, $ c$ and $ d$ are the sides of a quadrilateral inscribed in a given circle. Prove that the product $ (ab + cd)(ac + bd)(ad + bc)$ acquires its maximum when the quadrilateral is a square.
2008 Baltic Way
Let $ AB$ be a diameter of a circle $ S$, and let $ L$ be the tangent at $ A$. Furthermore, let $ c$ be a fixed, positive real, and consider all pairs of points $ X$ and $ Y$ lying on $ L$, on opposite sides of $ A$, such that $ |AX|\cdot |AY| = c$. The lines $ BX$ and $ BY$ intersect $ S$ at points $ P$ and $ Q$, respectively. Show that all the lines $ PQ$ pass through a common point.
2008 Baltic Way
In a circle of diameter $ 1$, some chords are drawn. The sum of their lengths is greater than $ 19$. Prove that there is a diameter intersecting at least $ 7$ chords.
2008 Baltic Way
Let $ M$ be a point on $ BC$ and $ N$ be a point on $ AB$ such that $ AM$ and $ CN$ are angle bisectors of the triangle $ ABC$. Given that $ \frac {\angle BNM}{\angle MNC} = \frac {\angle BMN}{\angle NMA}$, prove that the triangle $ ABC$ is isosceles.