Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Ukraine Team Selection Test
The $A$-excircle of the triangle $ABC$ touches the side $BC$ at point $K$. The circumcircles of triangles $AKB$ and $AKC$ intersect for the second time with the bisector of angle $A$ at points $X$ and $Y$ respectively. Let $M$ be the midpoint of $BC$. Prove that the circumcenter of triangle $XYM$ lies on $BC$.
2014 Ukraine Team Selection Test
Find all positive integers $n \ge  2$ such that equality  $i+j \equiv C_{n}^{i} +  C_{n}^{j}$ (mod $2$) is true for arbitrary $0 \le  i  \le j  \le n$.
2014 Ukraine Team Selection Test
Let $n \ge 3$ be an odd integer. Each cell is a $n \times  n$ board painted in yellow or blue. Let's call the sequence of cells $S_1, S_2,...,S_m$ path  if they are all the same color and the cells $S_i$ and $S_j$ have one in common an edge if and only if $|i - j| = 1$. Suppose that all yellow cells form a path and all the blue cells form a path. Prove that one of the two paths begins or ends at the center of the board.
2014 Ukraine Team Selection Test
For each natural $n \ge 4$, find the smallest natural number $k$ that satisfies following condition:

For an arbitrary arrangement of $k$ chips of two colors on $n\times n$ board, there exists a non-empty set such that all columns and rows contain even number ($0$ is also possible) of chips each color.
2014 Ukraine Team Selection Test
The quadrilateral $ABCD$ is inscribed in the circle $\omega$ with the center $O$. Suppose that the angles $B$ and $C$ are obtuse and lines $AD$ and $BC$ are not parallel. Lines $AB$ and $CD$ intersect at point $E$. Let $P$ and $R$ be the feet of the perpendiculars from the point $E$ on the lines $BC$ and $AD$ respectively. $Q$ is the intersection point of $EP$ and $AD, S$ is the intersection point of $ER$ and $BC$. Let K be the midpoint of the segment $QS$ . Prove that the points $E, K$, and $O$ are collinear.
2014 Ukraine Team Selection Test
Let $m, n$ be odd prime numbers.

Find all pairs of integers numbers $a, b$ for which the system of equations:

$x^m+y^m+z^m=a$,

$x^n+y^n+z^n=b$

has many solutions in integers $x, y, z$.
2014 Ukraine Team Selection Test
Find all positive integers $n \ge 4$ for which there are $n$ points in general position on the plane such that an arbitrary triangle with vertices belonging to the convex hull of these $n$ points, containing exactly one of $n - 3$ points inside remained.
2014 Ukraine Team Selection Test
Find all functions $f: R \to  R$ that satisfy the condition $(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)$ for arbitrary real $x, y, u, v$ such that $x + y = u + v$.
2014 Ukraine Team Selection Test
Prove that for an arbitrary prime $p \ge 3$ the number of positive integers $n$, for which $p | n! +1$ does not exceed  $cp^{2/3}$, where c is a constant that does not depend on $p$.
2013 Ukraine Team Selection Test
Let $ABC$ be an isosceles triangle $ABC$ with base $BC$ insribed in a circle. The segment $AD$ is the diameter of the circle, and point $P$ lies on the smaller arc $BD$. Line $DP$ intersects rays $AB$ and $AC$ at points $M$ and $N$, and the lines $BP$ and $CP$ intersects the line $AD$ at points $Q$ and $R$. Prove that the midpoint of the segment $MN$ lies on the circumscribed circle of triangle $PQR$.
2013 Ukraine Team Selection Test
The teacher reported to Peter an odd integer $m \le 2013$ and gave the guy a homework. Petrick should star the cells in the $2013 \times 2013$ table so to make the condition true: if there is an asterisk in some cell in the table, then or in row or column containing this cell should be no more than $m$ stars (including this one). Thus in each cell of the table the guy can put at most one star. The teacher promised Peter that his assessment would be just the number of stars that the guy will be able to place. What is the greatest number will the stars be able to place in the table Petrick?
2013 Ukraine Team Selection Test
For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.
2013 Ukraine Team Selection Test
For positive $x, y$, and $z$ that satisfy the condition $xyz = 1$, prove the inequality

$$\sqrt[3]{\frac{x+y}{2z}}+\sqrt[3]{\frac{y+z}{2x}}+\sqrt[3]{\frac{z+x}{2y}}\le \frac{5(x+y+z)+9}{8}$$
2013 Ukraine Team Selection Test
Six different points $A, B, C, D, E, F$  are marked on the plane, no four of them lie on one circle and no two segments with ends at these points lie on parallel lines. Let $P, Q,R$ be the points of intersection of the perpendicular bisectors  to pairs of segments $(AD, BE)$, $(BE, CF)$ ,$(CF, DA)$ respectively, and $P', Q' ,R'$ are points the intersection of the perpendicular bisectors to the pairs of segments $(AE, BD)$, $(BF, CE)$ , $(CA, DF)$ respectively. Show that $P \ne  P', Q \ne Q', R  \ne R'$, and prove that the lines $PP', QQ'$  and $RR'$  intersect at one point or are parallel.
2013 Ukraine Team Selection Test
$2013$ users have registered on the social network "Graph". Some users are friends, and friendship in "Graph" is mutual. It is known that among network users there are no three, each of whom would be friends. Find the biggest one possible number of pairs of friends in "Graph".
2013 Ukraine Team Selection Test
Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.
2013 Ukraine Team Selection Test
Determine all functions $f:\Bbb{R}\to\Bbb{R}$ such that $$  f^2(x+y)=f^2(x)+2f(xy)+f^2(y), $$  for all $x,y\in \Bbb{R}.$
2013 Ukraine Team Selection Test
Let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and rationals respectively.

a) Does there exist a partition of $\mathbb{Z}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?

b) Does there exist a partition of $\mathbb{Q}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?



Here $X+Y$ denotes the set $\{ x+y : x \in X, y \in Y \}$, for $X,Y \subseteq \mathbb{Z}$ and  for $X,Y \subseteq \mathbb{Q}$.
2013 Ukraine Team Selection Test
Specified natural number $a$. Prove that there are an infinite number of prime numbers $p$ such that for some natural $n$ the number $2^{2^n} + a$ is divisible by $p$.
2013 Ukraine Team Selection Test
$4026$ points were noted on the plane, not three of which lie on a straight line.

The $2013$ points are the vertices of a convex polygon, and the other $2013$ vertices are inside this polygon. It is allowed to paint each point in one of two colors. Coloring will be good if some pairs of dots can be combined segments with the following conditions:

$\bullet$ Each segment connects dots of the same color.

$\bullet$ No two drawn segments intersect at their inner points.

$\bullet$ For an arbitrary pair of dots of the same color, there is a path along the lines from one point to another.

Please note that the sides of the convex $2013$ rectangle are not automatically drawn segments, although some (or all) can be drawn as needed. Prove that the total number of good colors does not depend on the specific locations of the points and find that number.
2012 Ukraine Team Selection Test
Let $a, b, c$ be positive reals. Prove that $\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab}\ge 3 \sqrt{ab+bc+ca}$
2012 Ukraine Team Selection Test
$E$ is the intersection point of the diagonals of the cyclic quadrilateral, $ABCD, F$ is the intersection point of the lines $AB$ and $CD, M$ is the midpoint of the side $AB$, and $N$ is the midpoint of the side $CD$. The circles circumscribed around the triangles $ABE$ and $ACN$ intersect for the second time at point $K$. Prove that the points $F, K, M$ and $N$ lie on one circle.
2012 Ukraine Team Selection Test
A natural number $n$ is called perfect  if it is equal to the sum of all its natural divisors other than $n$. For example, the number $6$ is perfect because $6 = 1 + 2 + 3$. Find all even perfect numbers that can be given as the sum of two cubes positive integers.
2012 Ukraine Team Selection Test
Given an isosceles triangle $ABC$ ($AB = AC$), the inscribed circle $\omega$ touches its sides $AB$ and $AC$ at points $K$ and $L$, respectively. On the extension of the side of the base $BC$, towards $B$, an arbitrary point $M$. is chosen. Line $M$ intersects $\omega$ at the point $N$ for the second time, line $BN$ intersects the second point $\omega$ at the point $P$. On the line $PK$, there is a point $X$ such that $K$ lies between $P$ and $X$ and $KX = KM$. Determine the locus of the point $X$.
2012 Ukraine Team Selection Test
There are only two letters in the Mumu tribe alphabet: M and $U$. The word in the Mumu language is any sequence of letters $M$ and $U$, in which next to each letter $M$ there is a letter $U$ (for example, $UUU$ and $UMMUM$ are words and $MMU$ is not). Let  $f(m,u)$ denote the number of words in the Mumu language which have $m$ times the letter $M$ and $u$ times the letter $U$. Prove that $f (m, u) - f (2u - m + 1, u) = f (m, u - 1) - f (2u - m + 1, u - 1)$ for any  $u \ge 2,3 \le m \le 2u$.
2012 Ukraine Team Selection Test
For the positive integer $k$ we denote by the $a_n$ ,  the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.
2012 Ukraine Team Selection Test
Find all pairs of relatively prime integers $(x, y)$ that satisfy equality $2 (x^3 - x) = 5 (y^3 - y)$.
2012 Ukraine Team Selection Test
Call arrangement of $m$ number on the circle $m$-negative, if all numbers are equal to $-1$. On the first step Andrew chooses one number on  circle and multiplies it by $-1$. All other steps are similar: instead of the next number(clockwise) he writes its product with the number, written on the previous step. Prove that if $n$-negative arrangement in $k$ steps becomes $n$-negative again, then $(2^n - 1)$-negative after $(2^k - 1)$ steps becomes $(2^n - 1)$-negative again.
2012 Ukraine Team Selection Test
The inscribed circle $\omega$ of the triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$, respectively. Let $S$ be the intersection point of lines passing through points $B$ and $C$ and parallel to $A_1C_1$ and $A_1B_1$ respectively, $A_0$ be the foot of the perpendicular drawn from point $A_1$ on $B_1C_1$, $G_1$ be the centroid of triangle $A_1B_1C_1$, $P$ be the intersection point of the ray  $G_1A_0$ with $\omega$. Prove that points $S, A_1$, and $P$ lie on a straight line.
2012 Ukraine Team Selection Test
A unit square is cut by $n$ straight lines  . Prove that in at least one of these parts one can completely fit a square with side $\frac{1}{n+1}$



original wordingОдиничний квадрат розрізано  $n$ прямими на частини. Доведіть, що хоча б в одній з цих частин можна повністю розмістити квадрат зі стороною $\frac{1}{n+1}$



notesThe selection panel jury made a mistake because the solution known to it turned out to be incorrect. As it turned out, the assertion of the problem is still correct, although it cannot be proved by simple methods, see. article:

Keith Ball. Тhe plank problem for symmetric bodies  // Іпѵепііопез МаіЬешаІіеае. — 1991. — Ѵоі. 104, по. 1. — Р. 535-543.
2012 Ukraine Team Selection Test
Let $P$ be a polynomial with integer coefficients of degree $d$. For the set $A = \{ a_1, a_2, ..., a_k\}$ of positive integers we denote $S (A) = P (a_1) + P (a_2) + ... + P (a_k )$. The natural numbers $m, n$ are such that  $m ^{d+ 1} | n$.  Prove that the set $\{1, 2, ..., n\}$ can be subdivided into $m$ disjoint subsets $A_1, A_2, ..., A_m$ with the same number of elements such that $S (A_1) = S(A_2) = ... = S (A_m )$.
2012 Ukraine Team Selection Test
We shall call the triplet of numbers $a, b, c$ of the interval $[-1,1]$ qualitative  if these numbers satisfy the inequality $1 + 2abc\ge  a^2 + b^2 + c^2$. Prove that when the triples $a, b, c$, and $x, y, z$ are qualitative, then $ax, by, cz$ is also qualitative.
2011 Ukraine Team Selection Test
Given a right $ n $ -angle $ {{A} _ {1}} {{A} _ {2}} \ldots {{A} _ {n}} $, $n \ge 4 $, and a point $ M $ inside it. Prove the inequality $$\sin (\angle {{A} _ {1}} M {{A} _ {2}}) + \sin (\angle {{A} _ {2}} M {{A} _ {3}} ) + \ldots + \sin (\angle {{A} _ {n}} M {{A} _ {1}})> \sin \frac{2 \pi}{n} + (n-2) sin \frac{\pi}{n}$$
2011 Ukraine Team Selection Test
Given a positive integer $ n> 2 $. Prove that there exists a natural $ K $ such that for all integers $ k \ge K $ on the open interval $ ({{k} ^{n}}, \ {{(k + 1)} ^{n}}) $ there are $n$ different integers, the product of which is the $n$-th power of an integer.
2011 Ukraine Team Selection Test
Suppose an ordered set of $ ({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}}) $ real numbers, $n \ge 3 $. It is possible to replace the number $ {{a} _ {i}} $, $ i = \overline {2, \ n-1} $ by the number $ a_ {i} ^ {*} $ that $ {{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}} $. Let $ ({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}}) $ be the set with the largest sum of numbers that can be obtained from this, and $ ({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}}) $ is a similar set with the least amount.

For the odd $n \ge 3 $ and set $ (1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1) $ find the values of the expressions $ {{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}} $ and $ {{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}} $.
2011 Ukraine Team Selection Test
Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ $$f\left( f(x)^2y \right) = x^3 f(xy).$$



Proposed by Thomas Huber, Switzerland
2011 Ukraine Team Selection Test
The circle $ \omega $ inscribed in triangle $ABC$ touches its sides $AB, BC, CA$ at points $K, L, M$ respectively. In the arc $KL$ of the circle $ \omega $ that does not contain the point $M$, we select point $S$. Denote by $P, Q, R, T$ the intersection points of straight $AS$ and $KM, ML$ and $SC, LP$ and $KQ, AQ$ and $PC$ respectively. It turned out that the points $R, S$ and $M$ are collinear. Prove that the point $T$ also lies on the line $SM$.
2011 Ukraine Team Selection Test
Is there an increasing sequence of integers $ 0 = {{a} _{0}} <{{a} _{1}} <{{a} _{2}} <\ldots $ for which the following two conditions are satisfied simultaneously:

1) any natural number can be given as $ {{a} _{i}} + {{a} _{j}} $ for some (possibly equal) $ i \ge 0 $, $ j \ge 0$ ,

2) $ {{a} _ {n}}> \tfrac {{{n} ^ {2}}} {16} $ for all natural $ n $?
2011 Ukraine Team Selection Test
Inside the inscribed quadrilateral $ ABCD $, a point $ P $ is marked such that $ \angle PBC = \angle PDA $, $ \angle PCB = \angle PAD $. Prove that there exists a circle that touches the straight lines $ AB $ and $ CD $, as well as the circles circumscribed by the triangles $ ABP $ and $ CDP $.
2011 Ukraine Team Selection Test
Let $ H $ be the point of intersection of the altitudes $ AP $ and $ CQ $ of the acute-angled triangle $ABC$. The points $ E $ and $ F $ are marked on the median $ BM $ such that $ \angle APE = \angle BAC $, $ \angle CQF = \angle BCA $, with point $ E $ lying inside the triangle $APB$ and point $ F $ is inside the triangle $CQB$. Prove that the lines $AE, CF$, and $BH$ intersect at one point.
2011 Ukraine Team Selection Test
Let $ P (x) $ and $ Q (x) $ be polynomials with real coefficients such that $ P (0)> 0 $ and all coefficients of the polynomial $ S (x) = P (x) \cdot Q (x) $ are integers. Prove that for any positive $ x $ the inequality holds:

$$S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})). $$
2011 Ukraine Team Selection Test
Let $ n $ be a natural number. Consider all permutations $ ({{a} _ {1}}, \ \ldots, \ {{a} _ {2n}}) $ of the first $ 2n $ natural numbers such that the numbers $ | {{a} _ {i +1}} - {{a} _ {i}} |, \ i = 1, \ \ldots, \ 2n-1, $ are pairwise different. Prove that $ {{a} _ {1}} - {{a} _ {2n}} = n $ if and only if $ 1 \le {{a} _ {2k}} \le  n $ for all $ k = 1, \ \ldots, \ n $.
2010 Ukraine Team Selection Test
There are $2010$ red cards and $2010$ white cards. All of these $4020$ cards are shuffled and dealt in two randomly to each of the $2010$ round table players. The game consists of several rounds, each of which players simultaneously hand over cards to each other according to the following rules. If a player holds at least one red card, he passes one red card to the player sitting to his left, otherwise he transfers one white card to the left. The game ends after the round when each player has one red card and one white card. Determine as many rounds as possible.
2010 Ukraine Team Selection Test
Let $ABCD$ be a quadrilateral inscribled in a circle with the center $O, P$ be the point of intersection of the diagonals $AC$ and $BD$, $BC\nparallel AD$. Rays $AB$ and $DC$ intersect at the point $E$. The circle with center $I$ inscribed in the triangle $EBC$ touches $BC$ at point $T_1$. The $E$-excircle with center $J$ in the triangle $EAD$ touches the side $AD$ at the point T$_2$. Line $IT_1$ and $JT_2$ intersect at $Q$. Prove that the points $O, P$, and $Q$  lie on a straight line.
2010 Ukraine Team Selection Test
For the nonnegative numbers $a, b, c$ prove the inequality:

$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge \frac52$$
2010 Ukraine Team Selection Test
Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that  the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$.
2010 Ukraine Team Selection Test
Denote in the triangle $ABC$ by $h$ the length of the height drawn from vertex $A$, and by $\alpha = \angle BAC$. Prove that the inequality $AB + AC \ge  BC \cdot \cos \alpha   + 2h \cdot \sin \alpha$ . Are there triangles for which this inequality turns into equality?
2010 Ukraine Team Selection Test
Consider an infinite sequence of positive integers in which each positive integer occurs exactly once. Let $\{a_n\}, n\ge  1$ be such a sequence. We call it consistent  if, for an arbitrary natural $k$ and every natural $n ,m$ such that $a_n <a_m$, the inequality $a_{kn} <a _{km}$ also holds. For example, the sequence $a_n = n$ is consistent .

a) Prove that there are consistent sequences other than $a_n = n$.

b) Are there consistent sequences for which $a_n \ne n, n\ge  2$ ?

c) Are there consistent sequences for which $a n \ne n, n\ge  1$ ?
2010 Ukraine Team Selection Test
Let $ABC$ be the triangle in which $AB> AC$. Circle $\omega_a$ touches the segment of the $BC$ at point $D$, the extension of the segment $AB$ towards point $B$ at the point $F$, and the extension of the segment $AC$ towards point $C$ at the point $E$. The ray $AD$ intersects circle $\omega_a$ for second time at point $M$. Denote the circle circumscribed around the triangle $CDM$ by $\omega$. Circle $\omega$ intersects the segment $DF$ at N. Prove that $FN > ND$.
2010 Ukraine Team Selection Test
Is there a positive integer $n$ for which the following holds:

for an arbitrary rational $r$ there exists an integer $b$ and non-zero integers $a _1, a_2, ..., a_n$ such that $r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}$ ?
2009 Ukraine Team Selection Test
Let $ a$, $ b$, $ c$ are sides of a triangle. Find the least possible value $ k$ such that the following inequality always holds:

$ \left|\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}\right|<k$

(Vitaly Lishunov)
2009 Ukraine Team Selection Test
Let $S$ be a set consisting of $n$ elements, $F$ a set of subsets of $S$ consisting of $2^{n-1}$ subsets such that every three such subsets have a non-empty intersection.

a) Show that the intersection of all subsets of $F$ is not empty.

b) If you replace the number of sets from  $2^{n-1}$ with  $2^{n-1}-1$, will the previous answer change?
2009 Ukraine Team Selection Test
Let $n$ be some positive integer. Find all functions $f:{{R}^{+}}\to R$ (i.e., functions defined by the set of all positive real numbers with real values) for which equality holds $f\left( {{x}^{n+1}}+ {{y}^{n+1}} \right)={{x}^{n}}f\left( x \right)+{{y}^{n}}f\left( y \right)$ for any positive real numbers $x, y$
2009 Ukraine Team Selection Test
Let $A,B,C,D,E$ be consecutive points on a circle with center $O$ such that $AC=BD=CE=DO$. Let $H_1,H_2,H_3$ be the orthocenters  triangles $ACD,BCD,BCE$  respectively. Prove that the triangle $H_1H_2H_3$  is right.
2009 Ukraine Team Selection Test
Find all odd prime numbers $p$ for which there exists a natural number $g$ for which the sets $$A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}$$and $$B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}$$are equal.
2009 Ukraine Team Selection Test
Two circles $\gamma_1, \gamma_2$ are given, with centers at points $O_1, O_2$ respectively. Select a point $K$ on circle  $\gamma_2$ and construct two circles,  one $\gamma_3$ that touches circle $\gamma_2$ at   point $K$ and circle $\gamma_1$ at a point $A$, and the other $\gamma_4$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $B$. Prove that, regardless of the choice of  point K on circle $\gamma_2$, all lines $AB$ pass through a fixed point of the plane.
2009 Ukraine Team Selection Test
Suppose that integers are given $m <n $. Consider a spreadsheet of size $n \times n $, whose cells arbitrarily record all integers from $1 $ to ${{n} ^ {2}} $. Each row of the table is colored in yellow $m$ the largest elements. Similarly, the blue colors the $m$ of the largest elements in each column. Find the smallest number of cells that are colored yellow and blue at a time
2009 Ukraine Team Selection Test
Denote an acute-angle $\vartriangle ABC $ with sides $a, b, c $ respectively by ${{H}_{a}}, {{H}_{b}}, {{H}_{c}} $ the feet of altitudes ${{h}_{a}}, {{h}_{b}}, {{h}_{c}} $. Prove the inequality:

$$\frac {h_ {a} ^{2}} {{{a} ^{2}} - CH_ {a} ^{2}} + \frac{h_{b} ^{2}} {{{ b}^{2}} - AH_{b} ^{2}} + \frac{h_{c}^{2}}{{{c}^{2}} - BH_{c}^{2}} \ge 3 $$

(Dmitry Petrovsky)
2008 Ukraine Team Selection Test
There is a row that consists of digits from $ 0$ to $ 9$ and Ukrainian letters (there are $ 33$ of them) with following properties: there aren’t two distinct digits or letters $ a_i$, $ a_j$ such that $ a_i > a_j$ and $ i < j$ (if $ a_i$, $ a_j$ are letters $ a_i > a_j$ means that $ a_i$ has greater then $ a_j$ position in alphabet) and there aren’t two equal consecutive symbols or two equal symbols having exactly one symbol between them. Find the greatest possible number of symbols in such row.
2008 Ukraine Team Selection Test
For positive $ a, b, c, d$ prove that

$ (a + b)(b + c)(c + d)(d + a)(1 + \sqrt [4]{abcd})^{4}\geq16abcd(1 + a)(1 + b)(1 + c)(1 + d)$
2008 Ukraine Team Selection Test
Two circles $ \omega_1$ and $ \omega_2$ tangents internally in point $ P$. On their common tangent points $ A$, $ B$ are chosen such that $ P$ lies between $ A$ and $ B$. Let $ C$ and $ D$ be the intersection points of tangent from $ A$ to $ \omega_1$, tangent from $ B$ to $ \omega_2$ and tangent from $ A$ to $ \omega_2$, tangent from $ B$ to $ \omega_1$, respectively. Prove that $ CA + CB = DA + DB$.
2008 Ukraine Team Selection Test
Find all functions $ f: \mathbb{R}^{ + }\to\mathbb{R}^{ + }$ satisfying $ f\left(x + f\left(y\right)\right) = f\left(x + y\right) + f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ + }$ denotes the set of all positive reals.



Proposed by Paisan Nakmahachalasint, Thailand
2008 Ukraine Team Selection Test
Prove that there exist infinitely many pairs $ (a, b)$ of natural numbers not equal to $ 1$ such that $ b^b +a$ is divisible by $ a^a +b$.