Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Saint Petersburg Mathematical Olympiad
There is secret society with $2011$ members. Every member has bank account with integer balance ( can be negative).  Sometimes some member give one dollar to every his friend. It is known, that after some such moves members can redistribute their money  arbitrarily. Prove, that there are exactly $2010$ pairs of friends.
2011 Saint Petersburg Mathematical Olympiad
$n$ - some natural. We write on the board all such numbers $d$, that $d\leq 1000$ and $d|n+k$ for some $ 1\leq k \leq 1000$. Let $S(n)$ -sum of all written numbers. Prove , that $S(n)<10^6$ and $S(n)>10^6$ has infinitely many solutions.
2011 Saint Petersburg Mathematical Olympiad
In some city there are $2000000$ citizens. In every group of $2000$ citizens there are $3$ pairwise friends. Prove, that there are $4$ pairwise friends in city.
2011 Saint Petersburg Mathematical Olympiad
We have garland with $n$ lights. Some lights are on, some are off. In one move we can take some turned on light (only turned on) and turn off it and also change state of neigbour lights. We want to turn off all lights after some moves.. For what $n$ is it always possible?
2011 Saint Petersburg Mathematical Olympiad
$ABCD$ - convex quadrilateral.  $P$ is such point on $AC$ and inside $\triangle ABD$, that $$\angle ACD+\angle BDP = \angle ACB+ \angle DBP = 90-\angle BAD$$.

Prove that $\angle BAD+ \angle BCD =90$ or $\angle BDA + \angle CAB = 90$
2011 Saint Petersburg Mathematical Olympiad
$a,b$ are naturals and $$a \times GCD(a,b)+b \times LCM(a,b)<2.5 ab$$. Prove that $b|a$
2011 Saint Petersburg Mathematical Olympiad
Point $D$ is inside $\triangle ABC$ and $AD=DC$. $BD$ intersect $AC$ in $E$. $\frac{BD}{BE}=\frac{AE}{EC}$. Prove, that $BE=BC$
2011 Saint Petersburg Mathematical Olympiad
$ABCD$ - convex quadrilateral. $\angle A+ \angle D=150, \angle B<150, \angle C<150$ Prove, that area $ABCD$ is greater than $\frac{1}{4}(AB*CD+AB*BC+BC*CD)$
2011 Saint Petersburg Mathematical Olympiad
There is infinite sequence of composite numbers $a_1,a_2,...,$ where $a_{n+1}=a_n-p_n+\frac{a_n}{p_n}$ ; $p_n$ is smallest prime divisor of $a_n$. It is known, that $37|a_n$ for every $n$.

Find possible values of $a_1$
2011 Saint Petersburg Mathematical Olympiad
Sasha and Serg plays next game with $100$-angled regular polygon . In the beggining Sasha set natural numbers in every angle. Then they make turn by turn, first turn is made by Serg. Serg turn is to take two opposite angles and add $1$ to its numbers. Sasha turn is to take two neigbour angles and add $1$ to its numbers. Serg want to maximize amount of odd numbers. What maximal number of odd numbers can  he get no matter how Sasha plays?
2010 Saint Petersburg Mathematical Olympiad
Solve in positives $$x^y=z,y^z=x,z^x=y$$
2010 Saint Petersburg Mathematical Olympiad
$ABC$ is triangle with $AB=BC$. $X,Y$ are midpoints of $AC$ and $AB$. $Z$  is base of perpendicular from $B$ to $CY$. Prove, that circumcenter of $XYZ$ lies on $AC$
2010 Saint Petersburg Mathematical Olympiad
There are $2009$ cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Minisrty destroys 10 free roads. Can Business create cyclic route without self-intersections  through exactly $75$ different cities?
2010 Saint Petersburg Mathematical Olympiad
Natural number $N$ is given. Let $p_N$ - biggest prime, that $ \leq N$. On every move we replace $N$ by $N-p_N$. We repeat this until we get $0$ or $1$. If we get $1$ then $N$ is called as good, else is bad. For example, $95$ is good because we get $95 \to 6 \to 1$.

Prove that among numbers from $1$ to $1000000$ there are between one quarter and half good numbers
2010 Saint Petersburg Mathematical Olympiad
$SABCD$ is quadrangular pyramid. Lateral faces are acute triangles with orthocenters lying in one plane. $ABCD$ is base of pyramid and $AC$ and $BD$ intersects at $P$, where $SP$ is height of pyramid. Prove that $AC \perp BD$
2010 Saint Petersburg Mathematical Olympiad
For positive numbers is true that $$ab+ac+bc=a+b+c$$Prove $$a+b+c+1 \geq 4abc$$
2010 Saint Petersburg Mathematical Olympiad
$600$ integer numbers from $[1,1000]$ colored in red. Natural segment $[n,k]$ is called yummy if  for every natural $t$ from  $[1,k-n]$ there are two red numbers $a,b$ from $[n,k]$ and $b-a=t$ .

Prove that there is yummy segment with $[a,b]$ with $b-a \geq 199$
2010 Saint Petersburg Mathematical Olympiad
$f(x)$ is square trinomial. Is it  always possible to find polynomial $g(x)$ with fourth degree, such that $f(g(x))=0$ has not roots?
2010 Saint Petersburg Mathematical Olympiad
There are $10$ consecutive 30-digit numbers. We write the biggest divisor for every number ( divisor is not equal number). Prove that some written numbers ends with same digit.
2010 Saint Petersburg Mathematical Olympiad
$M,N$ are midpoints of $AB$ and $CD$ for convex quadrilateral $ABCD$. Points $X$ and $Y$ are on $ AD$ and $BC$ and $XD=3AX,YC=3BY$. $\angle MXA=\angle MYB = 90$.

Prove that $\angle XMN=\angle ABC$
2010 Saint Petersburg Mathematical Olympiad
There are $2010$ cities in country, and $3$ roads go from every city. President and Prime Minister play next game.

They sell roads by turn to one of $3$ companies( one road is one turn). President will win, if three roads from some city are sold to different companies.

Who will win?
2010 Saint Petersburg Mathematical Olympiad
Natural number $N$ is given. Let $p_N$ - biggest prime, that $ \leq N$. On every move we replace $N$ by $N-p_N$. We repeat this until we get $0$ or $1$. Prove that exists such number $N$, that we need exactly $1000$ turns to make $0$
2010 Saint Petersburg Mathematical Olympiad
$200 \times 200$ square is colored in chess order. In one move we can take every $2 \times 3$ rectangle and change color of all its cells. Can we make all cells of square in same color ?
2010 Saint Petersburg Mathematical Olympiad
Chess king is standing in some square of chessboard. Every sunday it is moved to one square by diagonal, and every another day it is moved to one square by horisontal or vertical. What maximal numbers of moves can be made ?
2010 Saint Petersburg Mathematical Olympiad
$a$ is irrational , but $a$ and $a^3-6a$ are roots of square polynomial with integer coefficients.Find $a$
2010 Saint Petersburg Mathematical Olympiad
$A$ -is $20$-digit number. We write $101$ numbers $A$ then erase last $11$ digits. Prove that this $2009$-digit number can not be degree of $2$
2010 Saint Petersburg Mathematical Olympiad
There are $2010$ cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Ministry destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly $11$ different cities?
2010 Saint Petersburg Mathematical Olympiad
For positive is true $$\frac{3}{abc} \geq a+b+c$$Prove $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq a+b+c$$
2010 Saint Petersburg Mathematical Olympiad
Incircle  of $ABC$ tangent $AB,AC,BC$ in $C_1,B_1,A_1$. $AA_1$ intersect incircle in $E$. $N$ is midpoint $B_1A_1$. $M$ is symmetric to $N$ relatively $AA_1$. Prove that $\angle EMC= 90$
2009 Saint Petersburg Mathematical Olympiad
$f(x)=ax^2+bx+c;a,b,c$ are reals.

$M=\{f(2n)|n \text{ is integer}\},N=\{f(2n+1)|n \text{ is integer}\}$ Prove that $M=N$ or $M \cap N = \O $
2009 Saint Petersburg Mathematical Olympiad
$[x,y]-[x,z]=y-z$ and $x \neq y \neq  z \neq x$

Prove, that $x|y,x|z$
2009 Saint Petersburg Mathematical Olympiad
Streets of Moscow are some circles (rings) with common center $O$ and some straight lines from center $O$ to external ring. Point $A,B$ - two crossroads on external ring. Three friends want to move from $A$ to $B$. Dima goes by external ring, Kostya goes from $A$ to $O$ then to $B$. Sergey says, that there is another way, that is shortest. Prove, that he is wrong.
2009 Saint Petersburg Mathematical Olympiad
From $2008 \times 2008$ square we remove one corner cell $1 \times 1$. Is number of ways to divide this figure to corners from $3$ cells odd or even ?
2009 Saint Petersburg Mathematical Olympiad
$O$ -circumcenter of $ABCD$. $AC$ and $BD$ intersect in $E$, $AD$ and $BC$ in $F$. $X,Y$ - midpoints of $AD$ and $BC$. $O_1$ -circumcenter of $EXY$. Prove that $OF \parallel O_1E$
2009 Saint Petersburg Mathematical Olympiad
Some cities in country are connected by road, and from every city goes $\geq 2008$ roads. Every road is colored in one of two colors. Prove, that exists cycle without self-intersections ,where $\geq 504$ roads and all roads are same color.
2009 Saint Petersburg Mathematical Olympiad
$f(x)=x^2+x$

$b_1,...,b_{10000}>0$ and $|b_{n+1}-f(b_n)|\leq \frac{1}{1000}$ for $n=1,...,9999$

Prove, that there is such $a_1>0$ that $a_{n+1}=f(a_n);n=1,...,9999$ and $|a_n-b_n|<\frac{1}{100}$
2009 Saint Petersburg Mathematical Olympiad
$x,y$ are naturals. $GCM(x^7,y^4)*GCM(x^8,y^5)=xy$ Prove that $xy$ is cube
2009 Saint Petersburg Mathematical Olympiad
$ABCD$ is convex quadrilateral with $AB=CD$. $AC$ and $BD$ intersect in $O$. $X,Y,Z,T$ are midpoints of $BC,AD,AC,BD$. Prove, that circumcenter of $OZT$ lies on $XY$.
2009 Saint Petersburg Mathematical Olympiad
$f(x),g(x),h(x)$ are square trinomials with discriminant, that equals $2$. And $f(x)+g(x),f(x)+h(x),g(x)+h(x)$ are square trinomials with discriminant, that equals $1$. Prove,that $f(x)+g(x)+h(x)$ has not roots.
2009 Saint Petersburg Mathematical Olympiad
Call a set of some cells in infinite chess field as board.  Set of rooks on the board call as awesome if no one rook can beat another, but every empty cell is under rook attack. There are awesome set with $2008$ rooks and with $2010$ rooks. Prove, that there are awesome set with $2009$ rooks.
2009 Saint Petersburg Mathematical Olympiad
$(x_n)$ is sequence, such that $x_{n+2}=|x_{n+1}|-x_n$. Prove, that it is periodic.
2009 Saint Petersburg Mathematical Olympiad
Points $Y,X$ lies on $AB,BC$ of $\triangle ABC$ and $X,Y,A,C$ are concyclic. $AX$ and $CY$ intersect in $O$. Points $M,N$ are midpoints of $AC$ and $XY$. Prove, that $BO$ is tangent to circumcircle of $\triangle MON$
2009 Saint Petersburg Mathematical Olympiad
$b,c$ are naturals. $b|c+1$ Prove that exists such natural  $x,y,z$ that $x+y=bz,xy=cz$
2009 Saint Petersburg Mathematical Olympiad
There are $40$ members of jury, that want to choose problem for contest. There are list with $30$ problems. They want to find such problem, that can be solved at least half members , but not all. Every member solved $26$ problems, and every two members solved different sets of problems.

Prove, that they can find problem for contest.
2009 Saint Petersburg Mathematical Olympiad
Points $A_1$ and $C_1$ are on $BC$ and $AB$ of acute-angled triangle $ABC$ . $AA_1$ and $CC_1$ intersect in $K$. Circumcircles of $AA_1B,CC_1B$ intersect in $P$ - incenter of $AKC$.

Prove, that $P$ - orthocenter of $ABC$
2009 Saint Petersburg Mathematical Olympiad
$ABC$ is acute-angled triangle. $AA_1,BB_1,CC_1$ are altitudes. $X,Y$ - midpoints of $AC_1,A_1C$. $XY=BB_1$.

Prove that one side of $ABC$ in $\sqrt{2}$ greater than other side.
2009 Saint Petersburg Mathematical Olympiad
Discriminants of square trinomials $f(x),g(x),h(x),f(x)+g(x),f(x)+h(x),g(x)+h(x)$ equals $1$.

Prove that $f(x)+h(x)+g(x) \equiv 0$
2008 Saint Petersburg Mathematical Olympiad
The graph $y=x^2+ax+b$ intersects any of the two axes at points $A$, $B$, and $C$. The incenter of triangle $ABC$ lies on the line $y=x$. Prove that $a+b+1=0$.
2008 Saint Petersburg Mathematical Olympiad
In a kingdom, there are roads open between some cities with lanes both ways, in such a way, that you can come from one city to another using those roads. The roads are toll, and the price for taking each road is distinct. A minister made a list of all routes that go through each city exactly once. The king marked the most expensive road in each of the routes and said to close all the roads that he marked at least once. After that, it became impossible to go from city $A$ to city $B$, from city $B$ to city $C$, and from city $C$ to city $A$. Prove that the kings order was followed incorrectly.
2008 Saint Petersburg Mathematical Olympiad
Pentagon $ABCDE$ has circle $S$ inscribed into it. Side $BC$ is tangent to $S$ at point $K$. If $AB=BC=CD$, prove that angle $EKB$ is a right angle.
2008 Saint Petersburg Mathematical Olympiad
The numbers $x_1,...x_{100}$ are written on a board so that $ x_1=\frac{1}{2}$ and for every $n$ from $1$ to $99$, $x_{n+1}=1-x_1x_2x_3*...*x_{100}$. Prove that $x_{100}>0.99$.
2008 Saint Petersburg Mathematical Olympiad
Given are distinct natural numbers $a$, $b$, and $c$. Prove that

$$ \gcd(ab+1, ac+1, bc+1)\le \frac{a+b+c}{3}$$
2008 Saint Petersburg Mathematical Olympiad
In cyclic quadrilateral $ABCD$ rays $AB$ and $DC$ intersect at point $E$, while segments $AC$ and $BD$ intersect at $F$. Point $P$ is on ray $EF$ such that angles $BPE$ and $CPE$ are congruent. Prove that angles $APB$ and $DPC$ are also equal.
2008 Saint Petersburg Mathematical Olympiad
A square with side $2008$ is broken into regions that are all squares with side $1$. In every region, either $0$ or $1$ is written, and the number of $1$'s and $0$'s is the same. The border between two of the regions is removed, and the numbers in each of them are also removed, while in the new region, their arithmetic mean is recorded. After several of those operations, there is only one square left, which is the big square itself. Prove that it is possible to perform these operations in such a way, that the final number in the big square is less than $\frac{1}{2^{10^6}}$.
2008 Saint Petersburg Mathematical Olympiad
Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point.



EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet.



Note: fresh translation
2008 Saint Petersburg Mathematical Olympiad
Point $O$ is the center of the circle into which quadrilateral $ABCD$ is inscribed. If angles $AOC$ and $BAD$ are both equal to $110$ degrees and angle $ABC$ is greater than angle $ADC$, prove that $AB+AD>CD$.



Fresh translation.
2008 Saint Petersburg Mathematical Olympiad
A wizard thinks of a number from $1$ to $n$. You can ask the wizard any number of yes/no questions about the number. The wizard must answer all those questions, but not necessarily in the respective order. What is the least number of questions that must be asked in order to know what the number is for sure. (In terms of $n$.)



Fresh translation.
2008 Saint Petersburg Mathematical Olympiad
A diagonal of a 100-gon is called good if it divides the 100-gon into two polygons each with an odd number of sides. A 100-gon was split into triangles with non-intersecting diagonals, exactly 49 of which are good. The triangles are colored into two colors such that no two triangles that border each other are colored with the same color. Prove that there is the same number of triangles colored with one color as with the other.



Fresh translation; slightly reworded.
2008 Saint Petersburg Mathematical Olympiad
In a sequence, $x_1=\frac{1}{2}$ and $x_{n+1}=1-x_1x_2x_3...x_n$ for $n\ge 1$. Prove that $0.99<x_{100}<0.991$.



Fresh translation. This problem may be similar to one of the 9th grade problems.
2008 Saint Petersburg Mathematical Olympiad
We color some cells in $10000 \times 10000$ square, such that every $10 \times 10$ square and every $1 \times 100$ line have at least one coloring cell. What minimum number of cells we should color ?
2008 Saint Petersburg Mathematical Olympiad
There are $2008$ trinomials $x^2-a_kx+b_k$ where $a_k$ and $b_k$ are all different numbers from set $(1,2,...,4016)$. These trinomials has not common real roots. We mark all real roots on the $Ox$-axis.

Prove, that distance between some two marked points is $\leq \frac{1}{250}$
2008 Saint Petersburg Mathematical Olympiad
There are $100$ numbers on circle, and no one number is divided by other. In same time for all numbers we make next operation:

If $(a,b)$ are two neighbors ($a$ is left neighbor) , then we write between $a,b$ number $\frac{a}{(a,b)}$ and erase $a,b$

This operation was repeated some times. What maximum number of $1$ we can receive ?



Example: If we have circle with $3$ numbers $4,5,6$ then after operation we receive circle with numbers $\frac{4}{(4,5)}=4,\frac{5}{(5,6)}=5, \frac{6}{(6,4)}=3$.
2008 Saint Petersburg Mathematical Olympiad
All faces of the tetrahedron $ABCD $ are acute-angled triangles.$AK$ and $AL$ -are altitudes in faces $ABC$ and $ABD$. Points $C,D,K,L$ lies on circle. Prove, that $AB \perp CD$
2008 Saint Petersburg Mathematical Olympiad
$a+b+c \leq 3000000$ and $a\neq b \neq c \neq a$ and $a,b,c$ are naturals.

Find maximum $GCD(ab+1,ac+1,bc+1)$
2008 Saint Petersburg Mathematical Olympiad
There are $10000$ cities in country, and roads between some cities. Every city has $<100$ roads. Every cycle route with odd number of road consists of $\geq 101 $ roads.

Prove that we can divide all cities in $100$ groups with $100$ cities, such that every road leads from one group to other.
2007 Serbia National Math Olympiad
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$.  The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively.  Let $J$ be the incenter of triangle $BCD$.  Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
2007 Serbia National Math Olympiad
Triangle $\Delta GRB$  is dissected into $25$ small triangles as shown. All vertices of these triangles are painted in three colors so that the following conditions are satisfied: Vertex $G$ is painted in green, vertex $R$ in red, and $B$ in blue; Each vertex on side $GR$ is either green or red, each vertex on $RB$ is either red or blue, and each vertex on $GB$ is either green or blue. The vertices inside the big triangle are arbitrarily colored.



Prove that, regardless of the way of coloring, at least one of the $25$ small triangles has vertices of three different colors.
2007 Serbia National Math Olympiad
Determine all pairs of natural numbers $(x; n)$ that satisfy the equation

$$x^{3}+2x+1 = 2^{n}.$$
2007 Serbia National Math Olympiad
Let $k$ be a natural number. For each function $f : \mathbb{N}\to \mathbb{N}$  define the sequence of functions $(f_{m})_{m\geq 1}$ by $f_{1}= f$ and $f_{m+1}= f \circ f_{m}$ for $m \geq 1$ . Function $f$ is called $k$-nice if for each $n \in\mathbb{N}: f_{k}(n) = f (n)^{k}$.



(a) For which $k$ does there exist an injective $k$-nice function $f$ ?



(b) For which $k$ does there exist a surjective $k$-nice function $f$ ?
2007 Serbia National Math Olympiad
In a scalene triangle $ABC , AD, BE , CF$ are the angle bisectors $(D \in BC , E \in AC , F \in AB)$. Points $K_{a}, K_{b}, K_{c}$ on the incircle of triangle $ABC$ are such that $DK_{a}, EK_{b}, FK_{c}$ are tangent to the incircle and $K_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB$. Let $A_{1}, B_{1}, C_{1}$ be the midpoints of sides $BC , CA, AB$ , respectively. Prove that the lines $A_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c}$ intersect on the incircle of triangle $ABC$.
2007 Serbia National Math Olympiad
Let $k$ be a given natural number. Prove that for any positive numbers $x; y; z$ with

the sum $1$ the following inequality holds:

$$\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}}\geq \frac{1}{7}.$$

When does equality occur?
2008 Serbia National Math Olympiad
Find all nonegative integers $ x,y,z$ such that $ 12^x+y^4=2008^z$
2008 Serbia National Math Olympiad
Triangle $ \triangle ABC$ is given. Points $ D$ i $ E$ are on line  $ AB$ such that  $ D - A - B - E, AD = AC$ and $ BE = BC$. Bisector of internal angles at $ A$ and $ B$ intersect $ BC,AC$ at $ P$ and $ Q$, and circumcircle of  $ ABC$ at  $ M$ and  $ N$. Line which connects  $ A$ with center of circumcircle of $ BME$ and line which connects $ B$ and center of circumcircle of $ AND$ intersect at $ X$. Prove that $ CX \perp PQ$.
2008 Serbia National Math Olympiad
Let $ a$, $ b$, $ c$ be positive real numbers such that $ a + b + c = 1$. Prove inequality:

$$ \frac{1}{bc + a + \frac{1}{a}} + \frac{1}{ac + b + \frac{1}{b}} + \frac{1}{ab + c + \frac{1}{c}} \leqslant \frac{27}{31}.$$
2008 Serbia National Math Olympiad
Each point of a plane is painted in one of three colors. Show that there exists a triangle such that:

$ (i)$ all three vertices of the triangle are of the same color;

$ (ii)$ the radius of the circumcircle of the triangle is $ 2008$;

$ (iii)$ one angle of the triangle is either two or three times greater than one of the other two angles.
2008 Serbia National Math Olympiad
The sequence $ (a_n)_{n\ge 1}$ is defined by $ a_1 = 3$, $ a_2 = 11$ and $ a_n = 4a_{n-1}-a_{n-2}$, for $ n \ge 3$. Prove that each term of this sequence is of the form $ a^2 + 2b^2$ for some natural numbers $ a$ and $ b$.
2008 Serbia National Math Olympiad
In a convex pentagon $ ABCDE$, let $ \angle EAB = \angle ABC = 120^{\circ}$, $ \angle ADB = 30^{\circ}$ and $ \angle CDE = 60^{\circ}$. Let $ AB = 1$. Prove that the area of the pentagon is less than $ \sqrt {3}$.
2009 Serbia National Math Olympiad
In a scalene triangle $ABC$, $\alpha$ and $\beta$ respectively denote the interior angles at vertixes $A$ and $B$. The bisectors of these two angles meet the opposite sides of the triangle at points $D$ and $E$, respectively. Prove that the acute angle between the lines $DE$ and $AB$ does not exceed $ \frac{ | \alpha - \beta |}{3}$ .

Proposed by Dusan Djukic
2009 Serbia National Math Olympiad
Find the smallest natural number which is a multiple of $2009$ and whose sum of (decimal) digits equals $2009$

Proposed by Milos Milosavljevic
2009 Serbia National Math Olympiad
Determine the largest positive integer $n$ for which there exist pairwise different sets $\mathbb{S}_1 , ..., \mathbb{S}_n$ with the following properties:

$1$) $|\mathbb{S}_i \cup \mathbb{S}_j | \leq 2004$ for any two indices $1 \leq i, j\leq n$, and

$2$) $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1,2,...,2008 \}$ for any $1 \leq i < j < k \leq n$

Proposed by Ivan Matic
2009 Serbia National Math Olympiad
Let $x$, $y$, $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$.

Prove that $$\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1$$.

When does equality occur?

Proposed by Marko Radovanovic
2009 Serbia National Math Olympiad
Triangle ABC has incircle w centered as S that touches the sides BC,CA and AB at P,Q and R respectively. AB isn't equal AC, the lines QR and BC intersects at point M, the circle that passes through points B and C touches the circle w at point N, circumcircle of triangle MNP intersects with line AP at L (L isn't equal to P). Then prove that S,L and M lie on the same line
2010 Serbia National Math Olympiad
Some of $n$ towns are connected by two-way airlines. There are $m$ airlines in total. For $i = 1, 2, \cdots, n$, let $d_i$ be the number of airlines going from town $i$. If $1\le d_i \le 2010$ for each $i = 1, 2,\cdots, 2010$, prove that

$$\displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n$$

Find all $n$ for which equality can be attained.



Proposed by Aleksandar Ilic
2010 Serbia National Math Olympiad
In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$.



Proposed by Marko Djikic
2010 Serbia National Math Olympiad
Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$,

$$a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots   + a^{1!} + 1.$$



Proposed by Milos Milosavljevic