Navvye/Olympiad-Problems-CSV huggingface dataset:
1998 Ukraine National Math Olympiad
Let $AB$ and $CD$ be diameters of a circle with center $O$. For a point $M$ on a shorter arc $CB$, lines $MA$ and $MD$ meet the chord $BC$ at points $P$ and $Q$ respectively. Prove that the sum of the areas of the triangles $CPM$ and $MQB$ equals the area of triangle $DPQ$.
1998 Ukraine National Math Olympiad
Baron Munchausen claims that he can accomodate an arbitrary set of guests in the rooms of his castle in such a way that the guests in each room either all know each other or all do not know each other. Is his claim true? (Assume that the rooms are large enough to accomodate any number of people, but that there are finitely many rooms.)
1998 Ukraine National Math Olympiad
Find all pairs of polynomials $f(x),g(x)$ that satisfy $f(xy)=f(x)+g(x)f(y)$ for all $x,y$.
1998 Ukraine National Math Olympiad
Solve the equation $\sqrt{1+\{2x\}}=\lfloor x^2\rfloor+2\lfloor x\rfloor+3$ (where $\{a\}=a-\lfloor a\rfloor$).
1998 Ukraine National Math Olympiad
The altitude $CD$ of triangle $ABC$ meets the bisector $BK$ of this triangle at $M$ and the altitude $KL$ of $\triangle BKC$ at $N$. The circumcircle of triangle $BKN$ meets the side $AB$ at point $P\ne B$. Prove that the triangle $KPM$ is isosceles.
1998 Ukraine National Math Olympiad
For any numbers $0<x,y,z\le1$ prove the inequality

$$\frac x{1+y+zx}+\frac y{1+z+xy}+\frac z{1+x+yz}\le\frac3{x+y+z}.$$
1998 Ukraine National Math Olympiad
Consider a function $f:[0,1]\to[0,1]$. Suppose that there is a real number $0<\lambda<1$ such that $f(\lambda)\notin\{0,\lambda\}$ and the equality

$$f(f(x)+y)=f(x)+f(y)$$holds whenever the function is defined on the arguments.



(a) Give an example of such a function.

(b) Prove that for some $x\in[0,1]$,

$$\underbrace{f(f(\cdots f(}_{19}x)\cdots))=\underbrace{f(f(\cdots f(}_{98}x)\cdots)).$$
1998 Ukraine National Math Olympiad
Find the number of real roots of the equation

$$\frac2\pi\arccos x=\sqrt{1-x^2}+1.$$
1998 Ukraine National Math Olympiad
The numbers $1,1998,1999$ are written on the board. In each step it is allowed to replace one of the numbers by its square decreased by three times the product of the other two numbers. Is it possible to obtain three numbers with the sum $0$ after several steps?
1998 Ukraine National Math Olympiad
Two spheres are externally tangent at point $P$. The segments $AB$ and $CD$ touch the spheres with $A$ and $C$ lying on the first sphere and $B$ and $D$ on the second. Let $M$ and $N$ be the projections of the midpoints of segments $AC$ and $BD$ on the line connecting the centers of the spheres. Prove that $PM=PN$.
1998 Ukraine National Math Olympiad
The sequence $(x_n)$ is given by $x_1=1$ and

$$x_{n+1}=\frac{n^2}{x_n}+\frac{x_n}{n^2}+2\enspace\text{for }n\ge1.$$(a) Prove that $x_{n+1}\ge x_n$ for all $n\ge4$.

(b) Prove that $\lfloor x_n\rfloor=n$ for all $n\ge4$.
1999 Ukraine National Mathematical Olympiad
Solve the system of equations $2|x|+|y|=1$, $\lfloor|x|\rfloor+\lfloor2|y|\rfloor=2$.
1999 Ukraine National Mathematical Olympiad
Is it possible to write numbers in the cells of a $7\times7$ board in such a way that the sum of numbers in every $2\times2$ or $3\times3$ square is divisible by $1999$, but the sum of all numbers in the board is not divisible by $1999$?
1999 Ukraine National Mathematical Olympiad
Is there a $2000$-digit number which is a perfect square and $1999$ of whose digits are fives?
1999 Ukraine National Mathematical Olympiad
Let $N$ be the point inside a rhombus $ABCD$ such that the triangle $BNC$ is equilateral. The bisector of $\angle ABN$ meets the diagonal $AC$ at $K$. Show that $BK=KN+ND$.
1999 Ukraine National Mathematical Olympiad
Consider the figure consisting of $19$ hexagonal cells, as shown in the picture. At the cell $A$, there is a piece that is allowed to move one cell up, up-right, or down-right. How many ways are there for the piece to reach the cell $B$, not passing through the cell $C$?
1999 Ukraine National Mathematical Olympiad
Describe the region in the coordinate plane defined by $|x^2+xy|\ge|x^2-xy|$.
1999 Ukraine National Mathematical Olympiad
Let $x$ and $y$ be positive real numbers with $(x-1)(y-1)\ge1$. Prove that for sides $a,b,c$ of an arbitrary triangle we have $a^2x+b^2y>c^2$.
1999 Ukraine National Mathematical Olympiad
Show that the number $9999999+1999000$ is composite.
1999 Ukraine National Mathematical Olympiad
The bisectors of angles $A,B,C$ of a triangle $ABC$ intersect the circumcircle of the triangle at $A_1,B_1,C_1$, respectively. Let $P$ be the intersection of the lines $B_1C_1$ and $AB$, and $Q$ be the intersection of the lines $B_1A_1$ and $BC$. Show how to construct the triangle $ABC$ by a ruler and a compass, given its circumcircle, points $P$ and $Q$, and the halfplane determined by $PQ$ in which point $B$ lies.
1999 Ukraine National Mathematical Olympiad
Solve the equation $\lfloor x\rfloor+\frac{1999}{\lfloor x\rfloor}=\{x\}+\frac{1999}{\{x\}}$.
1999 Ukraine National Mathematical Olympiad
Find all pairs $(k,l)$ of positive integers such that $\frac{k^l}{l^k}=\frac{k!}{l!}$.
1999 Ukraine National Mathematical Olympiad
Let $M$ be a fixed point inside a given circle. Two perpendicular chords $AC$ and $BD$ are drawn through $M$, and $K$ and $L$ are the midpoints of $AB$ and $CD$, respectively. Prove that the quantity $AB^2+CD^2-2KL^2$ is independent of the chords $AC$ and $BD$.
1999 Ukraine National Mathematical Olympiad
A sequence of natural numbers $(a_n)$ satisfies $a_{a_n}+a_n=2n$ for all $n\in\mathbb N$. Prove that $a_n=n$.
1999 Ukraine National Mathematical Olympiad
Solve the equation $\sin x\sin2x\sin3x+\cos x\cos2x\cos3x=1$.
1999 Ukraine National Mathematical Olympiad
Let $M$ be a point inside a triangle $ABC$. The line through $M$ parallel to $AC$ meets $AB$ at $N$ and $BC$ at $K$. The lines through $M$ parallel to $AB$ and $BC$ meet $AC$ at $D$ and $L$, respectively. Another line through $M$ intersects the sides $AB$ and $BC$ at $P$ and $R$ respectively such that $PM=MR$. Given that the area of $\triangle ABC$ is $S$ and that $\frac{CK}{CB}=a$, compute the area of $\triangle PQR$.
1999 Ukraine National Mathematical Olympiad
Two players alternately write integers on a blackboard as follows: the first player writes $a_1$ arbitrarily, then the second player writes $a_2$ arbitrarily, and thereafter a player writes a number that is equal to the sum of the two preceding numbers. The player after whose move the obtained sequence contains terms such that $a_i-a_j$ and $a_{i+1}-a_{j+1}~(i\ne j)$ are divisible by $1999$, wins the game. Which of the players has a winning strategy?
1999 Ukraine National Mathematical Olympiad
Evaluate

$$\lfloor\pi\rfloor+\left\lfloor\frac{\lfloor2\pi\rfloor}2\right\rfloor+\left\lfloor\frac{\lfloor3\pi\rfloor}3\right\rfloor+\ldots+\left\lfloor\frac{\lfloor1999\pi\rfloor}{1999}\right\rfloor.$$
1999 Ukraine National Mathematical Olympiad
Solve the equation $m^3-n^3=7mn+5$ in positive integers.
1999 Ukraine National Mathematical Olympiad
If $x_1$, $x_2$, ..., $x_6$ $\in $ $[0,1]$, prove that the cyclic sum of



$\frac{x_1^3}{x_2^5+x_3^5+x_4^5+x_5^5+x_6^5+5}$ is less than $\frac{3}{5}$.
1999 Ukraine National Mathematical Olympiad
Let $AA_1,BB_1,CC_1$ be the altitudes of an acute-angled triangle $ABC$, and let $O$ be an arbitrary interior point. Let $M,N,P,Q,R,S$ be the feet of the perpendiculars from $O$ to the lines $AA_1,BC,BB_1,CA,CC_1,AB$, respectively. Prove that the lines $MN,PQ,RS$ are concurrent.
1999 Ukraine National Mathematical Olympiad
Solve the equation

$$(\sin x)^{1998}+(\cos x)^{-1999}=(\cos x)^{1998}+(\sin x)^{-1999}.$$
1999 Ukraine National Mathematical Olympiad
Find all values of the parameter $k$ for which the system of inequalities

\begin{align*}

ky^2+4ky-2x+6k+3&\le0\\
kx^2-2y-2kx+3k-3&\le0

\end{align*}has a unique solution.
1999 Ukraine National Mathematical Olympiad
All faces of a parallelepiped $ABCDA_1B_1C_1D_1$ are rhombi, and their angles at $A$ are all equal to $\alpha$. Points $M,N,P,Q$ are selected on the edges $A_1B_1,DC,BC,A_1D_1$, respectively, such that $A_1M=BP$ and $DN=A_1Q$. Find the angle between the intersection lines of the plane $A_1BD$ with the planes $AMN$ and $APQ$.
1999 Ukraine National Mathematical Olympiad
Can the number (a)19991998 (b)19991999 be written in the form $ n^{4}+m^{3}-m $, where n, m are integers?
1999 Ukraine National Mathematical Olympiad
Find all functions $f:\mathbb R\to\mathbb R$ such that

$$f(xy)+f(xz)-f(x)f(yz)\ge1\qquad\text{for all }x,y,z.$$
1999 Ukraine National Mathematical Olympiad
Suppose that the function $f(x)=\tan(a_1x+1)+\ldots+\tan(a_{10}x+1)$ has the period $T>0$, where $a_1,\ldots,a_{10}$ are positive numbers. Prove that

$$T\ge\frac\pi{10}\min\left\{\frac1{a_1},\ldots,\frac1{a_{10}}\right\}.$$
2005 Ukraine National Mathematical Olympiad
Find all pairs of real numbers $ (x,y)$ such that:



$ \frac{x-2}{y}+\frac{5}{xy}=\frac{4-y}{x}-\frac{|y-2x|}{xy}.$
2005 Ukraine National Mathematical Olympiad
Let $ AD$ be the median of a triangle $ ABC.$ Suppose that $ \angle ADB=45^{\circ}$ and $ \angle ACB=30^{\circ}$. Compute $ \angle BAD.$
2005 Ukraine National Mathematical Olympiad
A $ 6 \times 6$ board is filled out with positive integers. Each move consists of selecting a square larger than $ 1 \times 1$, consisting of entire cells, and increasing all numbers inside the selected square by $ 1$. Is it always possible to perform several moves so as to reach a situation where all numbers on the board are divisible by $ 3$?
2005 Ukraine National Mathematical Olympiad
For which real numbers $ x>1$ is there a triangle with side lengths $ x^4+x^3+2x^2+x+1, 2x^3+x^2+2x+1,$ and $ x^4-1$?
2005 Ukraine National Mathematical Olympiad
Are there integers $ a,b,c,d,x,y,z,t$ such that each of the numbers:



$ |ay-bx|,|az-cx|,|at-dx|,|bz-cy|,|bt-dy|,|ct-dz|$



equals either $ 1$ or $ 2005$?
2005 Ukraine National Mathematical Olympiad
A convex quadrilateral $ ABCD$ with $ BC=CD$ and $ \angle CBA+\angle DAB>180^{\circ}$ is given. Suppose that $ W$ and $ Q$ are points on the sides $ BC$ and $ DC$ respectively (distinct from the vertices) such that $ AD=QD$ and $ WQ \parallel AD.$ Also suppose that $ AQ$ and $ BD$ intersect at a point $ M$ that is equidistant from the lines $ AD$ and $ BC$. Prove that angle $ \angle BWD=\angle ADW.$
2005 Ukraine National Mathematical Olympiad
Draw the locus of points $ M(x,y)$ in the Cartesian plane $ xOy$ satisfying $ (x^2-1)(|y|-1) \ge 0$.
2005 Ukraine National Mathematical Olympiad
Find all pairs of positive integers $ (m,n)$ such that $ \sqrt{m}+\frac{2005}{\sqrt{n}}=2006.$
2005 Ukraine National Mathematical Olympiad
On the plane are given $ n \ge 3$ points, not all on the same line. For any point $ M$ on the same plane, $ f(M)$ is defined to be the sum of the distances from $ M$ to these $ n$ points. Suppose that there is a point $ M_1$ such that $ f(M_1)\le f(M)$ for any point $ M$ on the plane. Prove that if a point $ M_2$ satisfies $ f(M_1)=f(M_2),$ then $ M_1 \equiv M_2.$
2005 Ukraine National Mathematical Olympiad
Mykolka the numismatist possesses $241$ coins, each worth an integer number of turgiks. The total value of the coins is $360$ turgiks. Is is necessarily true that the coins can be divided into three groups of equal total value ?  :maybe:
2005 Ukraine National Mathematical Olympiad
Can $ 1$ be written as a sum of $ 2005$ different terms of the form $ \frac{1}{3n-1},$ where $ n$ is a positive integer?
2005 Ukraine National Mathematical Olympiad
For every positive integer $ n$, prove the inequality:



$ \frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+...+\frac{n+2}{n!+(n+1)!+(n+2)!}<\frac{1}{2}.$
2005 Ukraine National Mathematical Olympiad
Under the rules of the "Sea battle" game (no two ships may have a common point), can the following sets of rectangular ships be arranged on a $ 10 \times 10$ square board:



$ (a)$ two ships $ 1 \times 4$, $ 4$ ships $ 1 \times 3$, $ 6$ ships $ 1 \times 2$, and $ 8$ ships $ 1 \times 1$;



$ (b)$ two ships $ 1 \times 4$, $ 4$ ships $ 1 \times 3$, $ 6$ ships $ 1 \times 2$, $ 6$ ships $ 1 \times 1$, and $ 1$ ship $ 2 \times 2$;



$ (c)$ two ships $ 1 \times 4$, $ 4$ ships $ 1 \times 3$, $ 6$ ships $ 1 \times 2$, $ 4$ ships $ 1 \times 1$, and $ 2$ ships $ 2 \times 2$?
2005 Ukraine National Mathematical Olympiad
Let $ AB$ and $ CD$ be two disjoint chords of a circle. A point $ E$, distinct from $ A$ and $ B$, is taken on the chord $ AB$. Consider the arc $ AB$ not containing $ C$ and $ D$. Using a ruler and a compass, construct a point $ F$ on this arc such that $ \frac{PE}{EQ}=\frac{1}{2}$, where $ P$ and $ Q$ are the intersection points of $ AB$ with the segments $ FC$ and $ FD$, respectively.
2005 Ukraine National Mathematical Olympiad
Let $ 0 \le \alpha, \beta, \gamma \le \frac{\pi}{2}$ satisfy the conditions



$ \sin \alpha+\sin \beta+\sin \gamma=1, \sin \alpha \cos 2\alpha+\sin \beta\cos 2\beta+\sin \gamma \cos 2\gamma=-1.$



Find all possible values of $ \sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma$.
2005 Ukraine National Mathematical Olympiad
Is it possible to divide a $ 10 \times 10$ board into:



$ (a)$ $ 4$ figures of type $ 1$ and $ 21$ figures of type $ 2$?

$ (b)$ $ 4$ figures of type $ 1$, $ 19$ figures of type $ 2$, and $ 2$ figures of type $ 3$?
2005 Ukraine National Mathematical Olympiad
An integer $ n>101$ is divisible by $ 101$. Suppose that every divisor $ d$ of $ n$ with $ 1<d<n$ equals the difference of two divisors of $ n$. Prove that $ n$ is divisble by $ 100$.
2005 Ukraine National Mathematical Olympiad
Points $ P$ and $ Q$ do not lie on the diagonal $ AC$ of a parallelogram $ ABCD$ and satisfy $ \angle ABP=\angle ADP, \angle CBQ=\angle CDQ.$ Prove that $ \angle PAQ=\angle PCQ.$
2005 Ukraine National Mathematical Olympiad
Find all functions $ f: \mathbb{R}^{+}\rightarrow \mathbb{R}$ that satisfy:



$ f(x)f(y)=f(xy)+2005 \left( \frac{1}{x}+\frac{1}{y}+2004 \right)$ for all $ x,y>0.$
2005 Ukraine National Mathematical Olympiad
If $ a,b,c$ are positive real numbers, prove the inequality:



$ \frac {a^2}{b} + \frac {b^3}{c^2} + \frac {c^4}{a^3} \ge - a + 2b + 2c.$



P.S. Thank you for the observation. I've already corrected it.
2005 Ukraine National Mathematical Olympiad
A point $ M$ is taken on the perpendicular bisector of the side $ AC$ of an acute-angled triangle $ ABC$ so that $ M$ and $ B$ are on the same side of $ AC$. If $ \angle BAC=\angle MCB$ and $ \angle ABC+\angle MBC=180^{\circ}$, find $ \angle BAC.$
2005 Ukraine National Mathematical Olympiad
On the plane are marked $ 2005$ points, no three of which are collinear. A line is drawn through any two of the points. Show that the points can be painted in two colors so that for any two points of the same color the number of the drawn lines separating them is even. (Two points are separated by a line if they lie in different open half-planes determined by the line).
2005 Ukraine National Mathematical Olympiad
Solve in $ \mathbb{R}$ the equation $ \left| x -\frac{\pi}{6} \right|+\left| x+\frac{\pi}{3} \right|=\arcsin \frac{x^3-x+2}{2}.$
2005 Ukraine National Mathematical Olympiad
The sum of positive real numbers $ a,b,c$ equals $ 1$. Prove that:



$ \sqrt{\frac{1}{a}-1}\sqrt{\frac{1}{b}-1}+\sqrt{\frac{1}{b}-1}\sqrt{\frac{1}{c}-1}+\sqrt{\frac{1}{c}-1}\sqrt{\frac{1}{a}-1} \ge 6.$
2005 Ukraine National Mathematical Olympiad
In an acute-angled triangle $ ABC$, $ \omega$ is the circumcircle and $ O$ its center, $ \omega _1$ the circumcircle of triangle $ AOC$, and $ OQ$ the diameter of $ \omega _1$. Let $ M$ and $ N$ be points on the lines $ AQ$ and $ AC$ respectively such that the quadrilateral $ AMBN$ is a parallelogram. Prove that the lines $ MN$ and $ BQ$ intersect on $ \omega _1$.
2005 Ukraine National Mathematical Olympiad
Find all monotone (not necessarily strictly) functions $ f: \mathbb{R}^{+}_0\rightarrow \mathbb{R}$ such that:



$ f(x+y)-f(x)-f(y)=f(xy+1)-f(xy)-f(1) \forall x,y \ge 0$;



$ f(3)+3f(1)=3f(2)+f(0).$
2005 Ukraine National Mathematical Olympiad
Find all positive integers $ n$ for which $ \cos (\pi \sqrt{n^2+n}) \ge 0.$
2005 Ukraine National Mathematical Olympiad
A polygon on a coordinate grid is built of $ 2005$ dominoes $ 1 \times 2$. What is the smallest number of sides of an even length such a polygon can have?
2005 Ukraine National Mathematical Olympiad
Prove that for any integers $ n \ge 2$ there is a set $ A_n$ of $ n$ distinct positive integers such that for any two distinct elements $ i,j \in A_n, |i-j|$ divides $ i^2+j^2.$
2005 Ukraine National Mathematical Olympiad
In space are marked $ 2005$ points, no four of which are in the same plane. A plane is drawn through any three points. Show that the points can be painted in two colors so that for any two points of the same color the number of the drawn planes separating them is odd. (Two points are separated by a plane if they lie in different open half-spaces determined by the plane).
2009 Ukraine National Mathematical Olympiad
Find all positive integer solutions of equation $n^3 - 2 = k! .$
2009 Ukraine National Mathematical Olympiad
There is convex $2009$-gon on the plane.



a) Find the greatest number of vertices of $2009$-gon such that no two forms the side of the polygon.



b) Find the greatest number of vertices of $2009$-gon such that among any three of them there is one that is not connected with other two by side.
2009 Ukraine National Mathematical Olympiad
On the party every boy gave $1$ candy to every girl and every girl gave $1$ candy to every boy. Then every boy ate $2$ candies and every girl ate $3$ candies. It is known that $\frac 14$ of all candies was eaten. Find the greatest possible number of children on the party.
2009 Ukraine National Mathematical Olympiad
In the triangle $ABC$ given that $\angle ABC = 120^\circ .$ The bisector of $\angle B$ meet $AC$ at $M$ and external bisector of $\angle BCA$ meet $AB$ at $P.$ Segments $MP$ and $BC$ intersects at $K$. Prove that $\angle AKM = \angle KPC .$
2009 Ukraine National Mathematical Olympiad
Let $a, b, c$ be integers satisfying $ab + bc + ca = 1.$ Prove that $(1+ a^2 )(1+ b^2 )(1+ c^2 )$ is a perfect square.
2009 Ukraine National Mathematical Olympiad
In acute-angled triangle $ABC,$ let $M$ be the midpoint of $BC$ and let $K$ be a point on side $AB.$ We know that $AM$ meet $CK$ at $F.$ Prove that if $AK = KF$ then $AB = CF .$
2009 Ukraine National Mathematical Olympiad
Given $2009 \times 4018$ rectangular board. Frame is a rectangle $n \times n$ or $n \times(n + 2)$ for $ ( n \geq 3 )$ without all cells which don’t have any common points with boundary of rectangle. Rectangles $1\times1,1\times 2,1\times 3$ and $ 2\times 4$ are also frames. Two players by turn paint all cells of some frame that has no painted cells yet. Player that can't make such move loses. Who has a winning strategy?
2009 Ukraine National Mathematical Olympiad
а) Prove that for any positive integer $n$ there exist a pair of positive integers $(m, k)$ such that

$${k + m^k + n^{m^k}} = 2009^n.$$

b) Prove that there are infinitely many positive integers $n$ for which there is only one such pair.
2009 Ukraine National Mathematical Olympiad
Build the set of points $( x, y )$ on coordinate plane, that satisfies equality:

$$ \sqrt{1-x^2}+\sqrt{1-y^2}=2-x^2-y^2.$$
2009 Ukraine National Mathematical Olympiad
In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$
2009 Ukraine National Mathematical Olympiad
Let $x \leq y \leq z \leq t$ be real numbers such that $xy + xz + xt + yz + yt + zt = 1.$



a) Prove that $xt < \frac 13,$



b) Find the least constant $C$ for which inequality $xt < C$ holds for all possible values $x$ and $t.$
2009 Ukraine National Mathematical Olympiad
Pairwise distinct real numbers $a, b, c$ satisfies the equality

$$a +\frac 1b =b + \frac 1c =c+\frac 1a.$$

Find all possible values of $abc .$
2009 Ukraine National Mathematical Olympiad
Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$
2009 Ukraine National Mathematical Olympiad
In the trapezoid $ABCD$ we know that $CD \perp BC, $ and $CD \perp AD .$ Circle $w$ with diameter $AB$ intersects $AD$ in points $A$ and $P,$ tangent from $P$ to $w$ intersects $CD$ at $M.$ The second tangent from $M$ to $w$ touches $w$ at $Q.$ Prove that midpoint of $CD$ lies on $BQ.$
2009 Ukraine National Mathematical Olympiad
Compare the number of distinct prime divisors of $200^2 \cdot 201^2 \cdot ... \cdot 900^2$ and $(200^2 -1)(201^2 -1)\cdot ... \cdot (900^2 -1) .$
2009 Ukraine National Mathematical Olympiad
There is a knight in the left down corner of $2009 \times 2009$ chessboard. The row and the column containing this corner are painted. The knight cannot move into painted cell and after its move new row and column that contains a square with knight become painted. Is it possible to paint all rows and columns of the chessboard?
2009 Ukraine National Mathematical Olympiad
Find all functions $f : \mathbb R \to \mathbb R$ such that

$$f\left(x+xy+f(y)\right)= \left( f(x)+\frac 12 \right) \left( f(y)+\frac 12 \right) \qquad \forall x,y \in \mathbb R.$$
2009 Ukraine National Mathematical Olympiad
Given a $n \times n$ square board. Two players by turn remove some side of unit square if this side is not a bound of $n \times n$ square board. The player lose if after his move $n \times n$ square board became broken into two parts. Who has a winning strategy?
2009 Ukraine National Mathematical Olympiad
Let $ABCD$ be a parallelogram with $\angle BAC = 45^\circ,$ and $AC > BD .$ Let $w_1$ and $w_2$ be two circles with diameters $AC$ and $DC,$ respectively. The circle $w_1$ intersects $AB$ at $E$ and the circle $w_2$ intersects $AC$ at $O$ and $C$, and $AD$ at $F.$ Find the ratio of areas of triangles $AOE$ and $COF$ if $AO = a,$ and $FO = b .$
2009 Ukraine National Mathematical Olympiad
Find all possible real values of $a$ for which the system of equations

$$\{\begin{array}{cc}x +y +z=0\\\text{ } \\ xy+yz+azx=0\end{array}$$

has exactly one solution.
2009 Ukraine National Mathematical Olympiad
Let $M = \{1, 2, 3, 4, 6, 8,12,16, 24, 48\} .$ Find out which of four-element subsets of $M$ are more: those with product of all elements greater than $2009$ or those with product of all elements less than $2009.$
2009 Ukraine National Mathematical Olympiad
In triangle $ABC$ let $M$ and $N$ be midpoints of $BC$ and $AC,$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PBC = \angle PCA .$ Prove that if $\angle PNA = \angle AMB,$ then $ABC$ is isosceles triangle.
2009 Ukraine National Mathematical Olympiad
Find all polynomials $P(x)$ with real coefficients such that for all pairwise distinct positive integers $x, y, z, t$ with $x^2 + y^2 + z^2 = 2t^2$ and $\gcd(x, y, z, t ) = 1,$ the following equality holds

$$2P^2(t ) + 2P(xy + yz + zx) = P^2(x + y + z) .$$



Note. $P^2(k)=\left( P(k) \right)^2.$
2009 Ukraine National Mathematical Olympiad
Solve the system of equations

$$\{\begin{array}{cc}x^3=2y^3+y-2\\ \text{ } \\ y^3=2z^3+z-2 \\ \text{ } \\ z^3 = 2x^3 +x -2\end{array}$$
2009 Ukraine National Mathematical Olympiad
Find all functions $f : \mathbb Z \to \mathbb Z$ such that

$$f (n |m|) + f (n(|m| +2)) = 2f (n(|m| +1)) \qquad \forall m,n \in \mathbb Z.$$



Note. $|x|$ denotes the absolute value of the integer $x.$
2009 Ukraine National Mathematical Olympiad
Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that

$$\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.$$
2009 Ukraine National Mathematical Olympiad
Let $G$ be a connected graph, with degree of all vertices not less then $m \geq 3$, such that there is no path through all vertices of $G$ being in every vertex exactly once. Find the least possible number of vertices of $G.$
2014 Ukraine National Mathematical Olympiad
Suppose that for real $x,y,z,t$ the following equalities hold:$\{x+y+z\}=\{y+z+t\}=\{z+t+x\}=\{t+x+y\}=1/4$.

Find all possible values of $\{x+y+z+t\}$.(Here$\{x\}=x-[x]$)