Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Baltic Way
The set $\{1, 2, . . . , 10\}$ is partitioned to three subsets $A, B$ and $C.$ For each subset the sum of its elements, the product of its elements and the sum of the digits of all its elements are calculated.

Is it possible that $A$ alone has the largest sum of elements, $B$ alone has the largest product of elements, and $C$ alone has the largest sum of digits?
2016 Baltic Way
Find all positive integers $n$ for which $$3x^n + n(x + 2) - 3 \geq  nx^2$$holds for all real numbers $x.$
2016 Baltic Way
Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R:$

i) $f(ax) = a^2f(x)$ and

ii) $f(f(x)) = a f(x).$
2016 Baltic Way
Find all quadruples $(a, b, c, d)$ of real numbers that simultaneously satisfy the following equations:

$$\begin{cases} a^3 + c^3 = 2 \\ a^2b + c^2d = 0 \\ b^3 + d^3 = 1 \\ ab^2 + cd^2 = -6.\end{cases}$$
2016 Baltic Way
Let $a_{0,1}, a_{0,2}, . . . , a_{0, 2016}$  be positive real numbers. For $n\geq  0$ and $1 \leq  k < 2016$ set $$a_{n+1,k} = a_{n,k} +\frac{1}{2a_{n,k+1}} \ \  \text{and} \ \  a_{n+1,2016} = a_{n,2016} +\frac{1}{2a_{n,1}}.$$Show that $\max_{1\leq k \leq 2016} a_{2016,k} > 44.$
2016 Baltic Way
Set $A$ consists of $2016$ positive integers. All prime divisors of these numbers are smaller than $30.$ Prove that there are four distinct numbers $a, b, c$ and $d$ in $A$ such that $abcd$ is a perfect square.
2016 Baltic Way
Does there exist a hexagon (not necessarily convex) with side lengths $1, 2, 3, 4, 5, 6$ (not necessarily in this order) that can be tiled with a) $31$ b) $32$ equilateral triangles with side length $1?$
2016 Baltic Way
Let $n$ numbers all equal to $1$ be written on a blackboard. A move consists of replacing two numbers on the board with two copies of their sum. It happens that after $h$ moves all $n$ numbers on the blackboard are equal to $m.$ Prove that $h \leq \frac{1}{2}n \log_2 m.$
2016 Baltic Way
A cube consists of $4^3$ unit cubes each containing an integer. At each move, you choose a unit cube and increase by $1$ all the integers in the neighbouring cubes having a face in common with the chosen cube. Is it possible to reach a position where all the $4^3$ integers are divisible by $3,$ no matter what the starting position is?
2016 Baltic Way
The Baltic Sea has $2016$ harbours. There are two-way ferry connections between some of them. It is impossible to make a sequence of direct voyages $C_1 - C_2 - ... - C_{1062}$ where all the harbours $C_1, . . . , C_{1062}$ are distinct. Prove that there exist two disjoint sets $A$ and $B$ of $477$ harbours each, such that there is no harbour in $A$ with a direct ferry connection to a harbour in $B.$
2016 Baltic Way
In triangle $ABC,$ the points $D$ and $E$ are the intersections of the angular bisectors from $C$ and $B$ with the sides $AB$ and $AC,$ respectively. Points $F$ and $G$ on the extensions of $AB$ and $AC$ beyond $B$ and $C,$ respectively, satisfy $BF = CG = BC.$ Prove that $F G \parallel DE.$
2016 Baltic Way
Let $ABCD$ be a convex quadrilateral with $AB = AD.$ Let $T$ be a point on the diagonal $AC$ such that $\angle ABT + \angle ADT = \angle BCD.$ Prove that $AT + AC \geq  AB + AD.$
2016 Baltic Way
Let $ABCD$ be a parallelogram such that $\angle BAD = 60^{\circ}.$ Let $K$ and $L$ be the midpoints of $BC$ and $CD,$ respectively. Assuming that $ABKL$ is a cyclic quadrilateral, find $\angle ABD.$
2016 Baltic Way
Consider triangles in the plane where each vertex has integer coordinates. Such a triangle can be legally transformed by moving one vertex parallel to the opposite side to a different point with integer coordinates. Show that if two triangles have the same area, then there exists a series of legal transformations that transforms one to the other.
2016 Baltic Way
Let $ABCD$ be a cyclic quadrilateral with $AB$ and $CD$ not parallel. Let $M$ be the midpoint of $CD.$ Let $P$ be a point inside $ABCD$ such that $P A = P B = CM.$ Prove that $AB, CD$ and the perpendicular bisector of $MP$ are concurrent.
2015 Baltic Way
For $n\geq 2$ , an equilateral triangle is divided into $n^2$ congruent smaller equilateral triangles. Detemine all ways in which real numbers can be assigned to the $\frac{(n+1)(n+2)}{2}$ vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the big triangle.
2015 Baltic Way
Let $n$ be a positive integer and let $a_1,\cdots ,a_n$ be real numbers satisfying $0\le a_i\le 1$ for $i=1,\cdots ,n.$ Prove the inequality $$(1-{a_i}^n)(1-{a_2}^n)\cdots (1-{a_n}^n)\le (1-a_1a_2\cdots a_n)^n.$$
2015 Baltic Way
Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying , for any real $x$ , the identy $$P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})$$
2015 Baltic Way
A family wears clothes of three colors: red,blue and green,with a separate,identical laundry bin for each color. At the beginning of the first week,all bins are empty.Each week,the family generates a total of $10 kg $ of laundry(the proportion of each color is subject to variation).The laundry is sorted by color and placed in the bins.Next,the heaviest bin(only one of them,if there are several that are heaviest)is emptied and its content swashed.What is the minimal possible storing capacity required of the laundry bins in order for them never to overflow?
2015 Baltic Way
Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the equation $$|x|f(y)+yf(x)=f(xy)+f(x^2)+f(f(y))$$for all real numbers $x$ and $y$.
2015 Baltic Way
Two players play the following game. At the outset there are two piles, containing $10,000$ and $20,000$ tokens,respectively . A move consists of removing any positive number of tokens from a single pile $or$ removing $x>0$ tokens from one pile and $y>0$ tokens from the other , where $x+y$ is divisible by $2015$. The player who can not make a move loses. Which player has a winning strategy
2015 Baltic Way
There are $100$ members in a ladies' club.Each lady has had tea (in private) with exactly $56$ of her lady friends.The Board,consisting of the  $50$ most distinguished ladies,have all had tea with one another.Prove that the entire club may be split into two groups in such a way that,with in each group,any lady has had tea with any other.
2015 Baltic Way
With inspiration drawn from the rectilinear network of streets in New York , the Manhattan distance between two points $(a,b)$ and $(c,d)$ in the plane is defined to be $$|a-c|+|b-d|$$Suppose only two distinct Manhattan distance occur between all pairs of distinct points of some point set. What is the maximal number of points in such a set?
2015 Baltic Way
Let $n>2$ be an integer. A deck contains $\frac{n(n-1)}{2}$ cards,numbered $$1,2,3,\cdots , \frac{n(n-1)}{2}$$Two cards form a magic pair if their numbers are consecutive , or if their numbers are $1$ and $\frac{n(n+1)}{2}$. For which $n$ is it possible to distribute the cards into $n$ stacks in such a manner that, among the cards in any two stacks , there is exactly one magic pair?
2015 Baltic Way
A subset $S$ of $ {1,2,...,n}$ is called balanced if for every $a $ from $S $ there exists some $ b $from $S$, $b\neq a$, such that $ \frac{(a+b)}{2}$  is in $S$ as well.

(a) Let $k > 1 $be an integer and let $n = 2k$. Show that every subset $ S$ of ${1,2,...,n} $ with $|S| > \frac{3n}{4}$  is balanced.

(b) Does there exist an $n =2k$, with $ k > 1 $ an integer, for which every subset $ S$ of ${1,2,...,n} $ with $ |S| >\frac{2n}{3} $ is balanced?
2015 Baltic Way
The diagonals of parallelogram $ABCD$ intersect at $E$ . The bisectors of $\angle DAE$ and $\angle EBC$ intersect at $F$. Assume $ECFD$ is a parellelogram . Determine the ratio $AB:AD$.
2015 Baltic Way
A circle passes through vertex $B$ of the triangle $ABC$, intersects its sides $ AB $and $BC$ at points $K$ and $L$, respectively, and touches the side $ AC$ at its midpoint $M$. The point $N$ on the arc $BL$ (which does not contain $K$) is such that $\angle LKN = \angle ACB$. Find $\angle BAC $ given that the triangle $CKN$ is equilateral.
2015 Baltic Way
Let $D$ be the footpoint of the altitude from $B$ in the triangle $ABC$ , where $AB=1$ . The incircle of triangle $BCD$ coincides with the centroid of triangle $ABC$. Find the lengths of $AC$ and $BC$.
2015 Baltic Way
In the non-isosceles triangle $ABC$ an altitude from $A$ meets side $BC$ in $D$ . Let $M$ be the midpoint of $BC$ and let $N$ be the reflection of $M$ in $D$ . The circumcirle of triangle $AMN$ intersects the side $AB$ in $P\ne A$ and the side $AC$ in $Q\ne A$ . Prove that $AN,BQ$ and $CP$ are concurrent.
2015 Baltic Way
In triangle $ABC$, the interior and exterior angle bisectors of $ \angle BAC$ intersect the line $BC$ in $D $ and $E$, respectively. Let $F$ be the second point of intersection of the line $AD$ with the circumcircle of the triangle $ ABC$. Let $O$ be the circumcentre of the triangle $ ABC $and let $D'$ be the reflection of $D$ in $O$. Prove that $ \angle D'FE =90.$
2015 Baltic Way
Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n\geq 2$ for which $$P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor$$
2015 Baltic Way
Find all positive integers $n$ for which $n^{n-1} - 1$ is divisible by $2^{2015}$, but not by $2^{2016}$.
2015 Baltic Way
Let $f(x)=x^n + a_{n-1}x^{n-1} + ...+ a_0 $ be a polynomial of degree $ n\ge 1 $ with $ n$ (not necessarily distinct) integer roots. Assume that there exist distinct primes $p_0,p_1,..,p_{n-1}$ such that $a_i > 1$ is a power of $p_i$, for all $ i=0,1,..,n-1$. Find all possible values of $ n$.
2015 Baltic Way
Three pairwairs distinct positive integers $a,b,c,$ with $gcd(a,b,c)=1$, satisfy $$a|(b-c)^2 ,b|(a-c)^2 , c|(a-b)^2$$Prove that there doesnt exist a non-degenerate triangle with side lengths $a,b,c.$
2015 Baltic Way
For any integer $n \ge2$, we define $ A_n$ to be the number of positive integers $ m$ with the following property: the distance from $n$ to the nearest multiple of $m$ is equal to the distance from $n^3$ to the nearest multiple of $ m$. Find all integers $n \ge 2 $ for which $ A_n$ is odd. (Note: The distance between two integers $ a$ and $b$ is defined as $|a -b|$.)
2014 Baltic Way
Show that $$\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot  . . . \cdot  \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .$$
2014 Baltic Way
Let $a_0, a_1, . . . , a_N$ be real numbers satisfying $a_0 = a_N = 0$ and $$a_{i+1} - 2a_i + a_{i-1} = a^2_i$$ for $i = 1, 2, . . . , N - 1.$ Prove that $a_i\leq 0$ for $i = 1, 2, . . . , N- 1.$
2014 Baltic Way
Positive real numbers $a, b, c$ satisfy $\frac{1}{a} +\frac{1}{b} +\frac{1}{c} = 3.$ Prove the inequality  $$\frac{1}{\sqrt{a^3+ b}}+\frac{1}{\sqrt{b^3 + c}}+\frac{1}{\sqrt{c^3 + a}}\leq \frac{3}{\sqrt{2}}.$$
2014 Baltic Way
Find all functions $f$ defined on all real numbers and taking real values such that $$f(f(y)) + f(x - y) = f(xf(y) - x),$$ for all real numbers $x, y.$
2014 Baltic Way
Given positive real numbers $a, b, c, d$ that satisfy equalities $$a^2 + d^2 - ad = b^2 + c^2 + bc \ \  \text{and} \ \  a^2 + b^2 = c^2 + d^2$$ find all possible values of the expression $\frac{ab+cd}{ad+bc}.$
2014 Baltic Way
In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?
2014 Baltic Way
Let $p_1, p_2, . . . , p_{30}$ be a permutation of the numbers $1, 2, . . . , 30.$ For how many permutations does the equality $\sum^{30}_{k=1}|p_k - k| = 450 $ hold?
2014 Baltic Way
Albert and Betty are playing the following game. There are $100$ blue balls in a red bowl and $100$ red balls in a blue bowl. In each turn a player must make one of the following moves:

a) Take two red balls from the blue bowl and put them in the red bowl.

b) Take two blue balls from the red bowl and put them in the blue bowl.

c) Take two balls of different colors from one bowl and throw the balls away.

They take alternate turns and Albert starts. The player who first takes the last red ball from the blue bowl or the last blue ball from the red bowl wins.

Determine who has a winning strategy.
2014 Baltic Way
What is the least posssible number of cells that can be marked on an $n \times n$ board such that for each $m >\frac{ n}{2}$ both diagonals of any $m \times m$  sub-board contain a marked cell?
2014 Baltic Way
In a country there are $100$ airports. Super-Air operates direct flights between some pairs of airports (in both directions). The traffic of an airport is the number of airports it has a direct Super-Air connection with. A new company, Concur-Air, establishes a direct flight between two airports if and only if the sum of their traffics is at least $100.$ It turns out that there exists a round-trip of Concur-Air flights that lands in every airport exactly once. Show that then there also exists a round-trip of Super-Air flights that lands in every airport exactly once.
2014 Baltic Way
Let $\Gamma$  be the circumcircle of an acute triangle $ABC.$ The perpendicular to $AB$ from $C$ meets $AB$ at $D$ and $\Gamma$ again at $E.$ The bisector of angle $C$ meets $AB$ at $F$ and $\Gamma$ again at $G.$ The line $GD$ meets $\Gamma$  again at $H$ and the line $HF$ meets $\Gamma$  again at $I.$ Prove that $AI = EB.$
2014 Baltic Way
Triangle $ABC$ is given. Let $M$ be the midpoint of the segment $AB$ and $T$ be the midpoint of the arc $BC$ not containing $A$ of the circumcircle of $ABC.$ The point $K$ inside the triangle $ABC$ is such that $MATK$ is an isosceles trapezoid with $AT\parallel MK.$ Show that $AK = KC.$
2014 Baltic Way
Let $ABCD$ be a square inscribed in a circle $\omega$  and let $P$ be a point on the shorter arc $AB$ of $\omega$. Let $CP\cap BD = R$ and $DP \cap  AC = S.$

Show that triangles $ARB$ and $DSR$ have equal areas.
2014 Baltic Way
Let $ABCD$ be a convex quadrilateral such that the line $BD$ bisects the angle $ABC.$ The circumcircle of triangle $ABC$ intersects the sides $AD$ and $CD$ in the points $P$ and $Q,$ respectively. The line through $D$ and parallel to $AC$ intersects the lines $BC$ and $BA$ at the points $R$ and $S,$ respectively. Prove that the points $P, Q, R$ and $S$ lie on a common circle.
2014 Baltic Way
The sum of the angles $A$ and $C$ of a convex quadrilateral $ABCD$ is less than $180^{\circ} .$ Prove that $$AB \cdot CD + AD \cdot  BC < AC(AB + AD).$$
2014 Baltic Way
Determine whether $712! + 1$ is a prime number.
2014 Baltic Way
Do there exist pairwise distinct rational numbers $x, y$ and $z$ such that $$\frac{1}{(x - y)^2}+\frac{1}{(y - z)^2}+\frac{1}{(z - x)^2}= 2014?$$
2014 Baltic Way
Let $p$ be a prime number, and let $n$ be a positive integer. Find the number of quadruples $(a_1, a_2, a_3, a_4)$ with $a_i\in \{0, 1, \ldots, p^n - 1\}$ for $i = 1, 2, 3, 4$, such that $$p^n \mid (a_1a_2 + a_3a_4 + 1).$$
2014 Baltic Way
Let $m$ and $n$ be relatively prime positive integers. Determine all possible values of $$\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).$$
2014 Baltic Way
Consider a sequence of positive integers $a_1, a_2, a_3, . . .$  such that for $k \geq 2$ we have $a_{k+1} =\frac{a_k + a_{k-1}}{2015^i},$ where $2015^i$ is the maximal power of $2015$ that divides $a_k + a_{k-1}.$ Prove that if this sequence is periodic then its period is divisible by $3.$
2013 Baltic Way
Let $n$ be a positive integer. Assume that $n$ numbers are to be chosen from the table



$\begin{array}{cccc}0 & 1 & \cdots & n-1\\ n & n+1 & \cdots & 2n-1\\ \vdots & \vdots & \ddots & \vdots\\(n-1)n & (n-1)n+1 & \cdots & n^2-1\end{array} $



with no two of them from the same row or the same column. Find the maximal value of the product of these $n$ numbers.
2013 Baltic Way
Let $k$ and $n$ be positive integers and let $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$ be distinct integers. A polynomial $P$ with integer coefficients satisfies

$$P(x_1)=P(x_2)= \cdots = P(x_k)=54$$

$$P(y_1)=P(y_2)= \cdots = P(y_n)=2013.$$



Determine the maximal value of $kn$.
2013 Baltic Way
Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that

$$f(xf(y)+y)+f(-f(x))=f(yf(x)-y)+y$$

for all $x,y\in\mathbb{R}$
2013 Baltic Way
Prove that the following inequality holds for all positive real numbers $x,y,z$:

$$\dfrac{x^3}{y^2+z^2}+\dfrac{y^3}{z^2+x^2}+\dfrac{z^3}{x^2+y^2}\ge \dfrac{x+y+z}{2}.$$
2013 Baltic Way
Numbers $0$ and $2013$ are written at two opposite vertices of a cube. Some real numbers are to be written at the remaining $6$ vertices of the cube. On each edge of the cube the difference between the numbers at its endpoints is written. When is the sum of squares of the numbers written on the edges minimal?
2013 Baltic Way
Santa Claus has at least $n$ gifts for $n$ children. For $i\in\{1,2, ... , n\}$, the $i$-th child considers $x_i > 0$ of these items to be desirable. Assume that

$$\dfrac{1}{x_1}+\cdots+\dfrac{1}{x_n}\le1.$$

Prove that Santa Claus can give each child a gift that this child likes.
2013 Baltic Way
A positive integer is written on a blackboard. Players $A$ and $B$ play the following game: in each move one has to choose a proper divisor $m$ of the number $n$ written on the blackboard ($1<m<n$) and replaces $n$ with $n-m$. Player $A$ makes the first move, then players move alternately. The player who can't make a move loses the game. For which starting numbers is there a winning strategy for player $B$?
2013 Baltic Way
There are $n$ rooms in a sauna, each has unlimited capacity. No room may be attended by a female and a male simultaneously. Moreover, males want to share a room only with males that they don't know and females want to share a room only with females that they know. Find the biggest number $k$ such that any $k$ couples can visit the sauna at the same time, given that two males know each other if and only if their wives know each other.
2013 Baltic Way
In a country there are $2014$ airports, no three of them lying on a line. Two airports are connected by a direct flight if and only if the line passing through them divides the country in two parts, each with $1006$ airports in it. Show that there are no two airports such that one can travel from the first to the second, visiting each of the $2014$ airports exactly once.
2013 Baltic Way
A white equilateral triangle is split into $n^2$ equal smaller triangles by lines that are parallel to the sides of the triangle. Denote a line of triangles to be all triangles that are placed between two adjacent parallel lines that forms the grid. In particular, a triangle in a corner is also considered to be a line of triangles.



We are to paint all triangles black by a sequence of operations of the following kind: choose a line of triangles that contains at least one white triangle and paint this line black (a possible situation with $n=6$ after four operations is shown in Figure 1; arrows show possible next operations in this situation). Find the smallest and largest possible number of operations.
2013 Baltic Way
In an acute triangle $ABC$ with $AC > AB$, let $D$ be the projection of $A$ on $BC$, and let $E$ and $F$ be the projections of $D$ on $AB$ and $AC$, respectively. Let $G$ be the intersection point of the lines $AD$ and $EF$. Let $H$ be the second intersection point of the line $AD$ and the circumcircle of triangle $ABC$. Prove that $$AG \cdot AH=AD^2$$
2013 Baltic Way
A trapezoid $ABCD$ with bases $AB$ and $CD$ is such that the circumcircle of the triangle $BCD$ intersects the line $AD$ in a point $E$, distinct from $A$ and $D$. Prove that the circumcircle oF the triangle $ABE$ is tangent to the line $BC$.
2013 Baltic Way
All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron.
2013 Baltic Way
Circles $\alpha$ and $\beta$ of the same radius intersect in two points, one of which is $P$. Denote by $A$ and $B$, respectively, the points diametrically opposite to $P$ on each of $\alpha$ and $\beta$. A third circle of the same radius passes through $P$ and intersects $\alpha$ and $\beta$ in the points $X$ and $Y$ , respectively. Show that the line $XY$ is parallel to the line $AB$.
2013 Baltic Way
Four circles in a plane have a common center. Their radii form a strictly increasing arithmetic progression. Prove that there is no square with each vertex lying on a different circle.
2013 Baltic Way
We call a positive integer $n$ delightful if there exists an integer $k$, $1 < k < n$, such that

$$1+2+\cdots+(k-1)=(k+1)+(k+2)+\cdots+n$$

Does there exist a delightful number $N$ satisfying the inequalities

$$2013^{2013}<\dfrac{N}{2013^{2013}}<2013^{2013}+4 ?$$
2013 Baltic Way
Let $c$ and $n > c$ be positive integers. Mary's teacher writes $n$ positive integers on a blackboard. Is it true that for all $n$ and $c$ Mary can always label the numbers written by the teacher by $a_1,\ldots, a_n$ in such an order that the cyclic product $(a_1-a_2)\cdot(a_2-a_3)\cdots(a_{n-1}-a_n)\cdot(a_n-a_1)$ would be congruent to either $0$ or $c$ modulo $n$?
2013 Baltic Way
Find all pairs $(x,y)$ of integers such that $y^3-1=x^4+x^2$.
2013 Baltic Way
Let $a_0$ be a positive integer and $a_n=5a_{n-1}+4$ for all $n\ge 1$. Can $a_0$ be chosen so that $a_{54}$ is a multiple of $2013$?
2013 Baltic Way
Find all polynomials $f$ with non-negative integer coefficients such that for all primes $p$ and positive integers $n$ there exist a prime $q$ and a positive integer $m$ such that $f(p^n)=q^m$.
2012 Baltic Way
The numbers from 1 to 360 are partitioned into 9 subsets of consecutive integers and the sums of the numbers in each subset are arranged in the cells of a $3 \times 3$ square.  Is it possible that the square turns out to be a magic square?



Remark: A magic square is a square in which the sums of the numbers in each row, in each column and in both diagonals are all equal.
2012 Baltic Way
Let $a$, $b$, $c$ be real numbers.  Prove that

$$ab + bc + ca + \max\{|a - b|, |b - c|, |c - a|\} \le 1 + \frac{1}{3} (a + b + c)^2.$$
2012 Baltic Way
(a) Show that the equation

$$\lfloor x \rfloor (x^2 + 1) = x^3,$$

where $\lfloor x \rfloor$ denotes the largest integer not larger than $x$, has exactly one real solution in each interval between consecutive positive integers.



(b) Show that none of the positive real solutions of this equation is rational.
2012 Baltic Way
Prove that for infinitely many pairs $(a,b)$ of integers the equation

$$x^{2012} = ax + b$$

has among its solutions two distinct real numbers whose product is 1.
2012 Baltic Way
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ for which

$$f(x + y) = f(x - y) + f(f(1 - xy))$$

holds for all real numbers $x$ and $y$.
2012 Baltic Way
There are 2012 lamps arranged on a table.  Two persons play the following game.  In each move the player flips the switch of one lamp, but he must never get back an arrangement of the lit lamps that has already been on the table.  A player who cannot move loses.  Which player has a winning strategy?
2012 Baltic Way
On a $2012 \times 2012$ board, some cells on the top-right to bottom-left diagonal are marked.  None of the marked cells is in a corner.  Integers are written in each cell of this board in the following way.  All the numbers in the cells along the upper and the left sides of the board are 1's.  All the numbers in the marked cells are 0's.  Each of the other cells contains a number that is equal to the sum of its upper neighbour and its left neighbour.  Prove that the number in the bottom right corner is not divisible by 2011.
2012 Baltic Way
A directed graph does not contain directed cycles.  The number of edges in any directed path does not exceed 99.  Prove that it is possible to colour the edges of the graph in 2 colours so that the number of edges in any single-coloured directed path in the graph will not exceed 9.
2012 Baltic Way
Zeroes are written in all cells of a $5 \times 5$ board.  We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it.  Is it possible to obtain the number 2012 in all cells simultaneously?