Navvye/Olympiad-Problems-CSV huggingface dataset:
2010 Serbia National Math Olympiad
Let $O$ be the circumcenter of triangle $ABC$. A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$. Prove that $\angle APQ = 2\angle CAP$.



Proposed by Dusan Djukic
2010 Serbia National Math Olympiad
An $n\times n$ table whose cells are numerated with numbers $1, 2,\cdots, n^2$ in some order is called Naissus if all products of $n$ numbers written in $n$ scattered cells give the same residue when divided by $n^2+1$. Does there exist a Naissus table for

$(a) n = 8;$

$(b) n = 10?$

($n$ cells are scattered if no two are in the same row or column.)



Proposed by Marko Djikic
2010 Serbia National Math Olympiad
Let $a_0$ and $a_n$ be different divisors of a natural number $m$, and $a_0, a_1, \ldots, a_n$ be a sequence of natural numbers such that it satisfies

$$a_{i+1} = |a_i\pm a_{i-1}|\text{  for  }0 < i < n$$

If $gcd(a_0,a_1,\ldots, a_n) = 1$, show that there exists a term of the sequence that is smaller than $\sqrt{m}$ .



Proposed by Dusan Djukic
2011 Serbia National Math Olympiad
Let $n \ge 2$ be integer. Let $a_0$, $a_1$, ... $a_n$ be sequence of positive reals such that:

$(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}$, for $k=1, 2, ..., n-1$.

Prove $a_n< \frac{1}{n-1}$.
2011 Serbia National Math Olympiad
Let $n$ be an odd positive integer such that both $\phi(n)$ and $\phi (n+1)$ are powers of two. Prove $n+1$ is power of two or $n=5$.
2011 Serbia National Math Olympiad
Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$. $D$  and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$, respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$, and $M$ is midpoint of $AB$. Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$.
2011 Serbia National Math Olympiad
On sides $AB, AC, BC$ are points $M, X, Y$, respectively, such that $AX=MX$; $BY=MY$. $K$, $L$ are midpoints of $AY$ and $BX$. $O$ is circumcenter of $ABC$, $O_1$, $O_2$ are symmetric with $O$ with respect to $K$ and $L$. Prove that $X, Y, O_1, O_2$ are concyclic.
2011 Serbia National Math Olympiad
Are there positive integers $a, b, c$ greater than $2011$ such that:

$(a+ \sqrt{b})^c=...2010,2011...$?
2011 Serbia National Math Olympiad
Set $T$ consists of $66$ points in plane, and $P$ consists of $16$ lines in plane. Pair $(A,l)$ is good if $A \in T$, $l \in P$ and $A \in l$. Prove that maximum number of good pairs is no greater than $159$, and prove that there exits configuration with exactly $159$ good pairs.
2012 Serbia National Math Olympiad
Let $ABCD$ be a parallelogram and $P$ be a point on diagonal $BD$ such that $\angle PCB=\angle ACD$. Circumcircle of triangle $ABD$ intersects line $AC$ at points $A$ and $E$. Prove that $$\angle AED=\angle PEB.$$
2012 Serbia National Math Olympiad
Find all natural numbers $a$ and $b$ such that $$a|b^2, \quad b|a^2 \mbox{  and  } a+1|b^2+1.$$
2012 Serbia National Math Olympiad
A fly and $k$ spiders are placed in some vertices of $2012 \times 2012$ lattice. One move consists of following: firstly, fly goes to some adjacent vertex or stays where it is and then every spider goes to some adjacent vertex or stays where it is (more than one spider can be in the same vertex). Spiders and fly knows where are the others all the time.

a) Find the smallest $k$ so that spiders can catch the fly in finite number of moves, regardless of their initial position.

b) Answer the same question for three-dimensional lattice $2012\times 2012\times 2012$.



(Vertices in lattice are adjacent if exactly one coordinate of one vertex is different from the same coordinate of the other vertex, and their difference is equal to $1$. Spider catches a fly if they are in the same vertex.)
2012 Serbia National Math Olympiad
Find all natural numbers $n$ for which there is a permutation $(p_1,p_2,...,p_n)$ of numbers $(1,2,...,n)$ such that sets $\{p_1 +1, p_2 + 2,..., p_n +n\}$ and $\{p_1-1, p_2-2,...,p_n -n\}$ are complete residue systems $\mod n$.
2012 Serbia National Math Olympiad
Let $\mathbb{K}$ be two-dimensional integer lattice. Is there a bijection $f:\mathbb{N} \rightarrow \mathbb{K}$, such that for every distinct $a,b,c \in \mathbb{N}$ we have: $$\gcd(a,b,c)>1 \Rightarrow f(a),f(b),f(c) \mbox{  are not colinear?  }$$
2012 Serbia National Math Olympiad
We are given $n>1$ piles of coins. There are two different types of coins: real and fake coins; they all look alike, but coins of the same type have the same mass, while the coins from different types have different masses. Coins that belong to the same pile are of the same type. We know the mass of real coin.



Find the minimal number of weightings on digital scale that we need in order to conclude: which piles consists of which type of coins and also the mass of fake coin.



(We assume that every pile consists from infinite number of coins.)
2013 Serbia National Math Olympiad
Let $k$ be a natural number. Bijection $f:\mathbb{Z} \rightarrow \mathbb{Z}$ has the following property: for any integers $i$ and $j$, $|i-j|\leq k$ implies $|f(i) - f(j)|\leq k$. Prove that for every $i,j\in \mathbb{Z}$ it stands: $$|f(i)-f(j)|= |i-j|.$$
2013 Serbia National Math Olympiad
For a natural number $n$, set $S_n$ is defined as: $$S_n = \left \{ {n\choose 

n}, {2n \choose n}, {3n\choose n},..., {n^2 \choose n} \right \}.$$



a) Prove that there are infinitely many composite numbers $n$, such that the set $S_n$ is not complete residue system mod $n$;



b) Prove that there are infinitely many composite numbers $n$, such that the set $S_n$ is complete residue system mod $n$.
2013 Serbia National Math Olympiad
Let $M$, $N$ and $P$ be midpoints of sides $BC, AC$ and $AB$, respectively, and let $O$ be circumcenter of acute-angled triangle $ABC$. Circumcircles of triangles $BOC$ and $MNP$ intersect at two different points $X$ and $Y$ inside of triangle $ABC$. Prove that $$\angle BAX=\angle CAY.$$
2013 Serbia National Math Olympiad
Determine all natural numbers $n$ for which there is a partition of $\{1,2,...,3n\}$ in $n$ pairwise disjoint subsets of the form $\{a,b,c\}$, such that numbers $b-a$ and $c-b$ are different numbers from the set $\{n-1, n, n+1\}$.
2013 Serbia National Math Olympiad
Let $A'$ and $B'$ be feet of altitudes from $A$ and $B$, respectively, in acute-angled triangle $ABC$ ($AC\not = BC$). Circle $k$ contains points $A'$ and $B'$ and touches segment $AB$ in $D$. If triangles $ADA'$ and $BDB'$ have the same area, prove that $$\angle A'DB'= \angle ACB.$$
2013 Serbia National Math Olympiad
Find the largest constant $K\in \mathbb{R}$ with the following property:

if $a_1,a_2,a_3,a_4>0$ are numbers satisfying $a_i^2 + a_j^2 + a_k^2 \geq 2 

(a_ia_j + a_ja_k + a_ka_i)$, for every $1\leq i<j<k\leq 4$, then $$a_1^2+a_2^2+a_3^2+a_4^2 \geq K

(a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4).$$
2014 Serbia National Math Olympiad
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$ hold:

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



Proposed by Dusan Djukic
2014 Serbia National Math Olympiad
On sides $BC$ and $AC$ of $\triangle ABC$ given are $D$ and $E$, respectively. Let $F$ ($F \neq C$) be a point of intersection of circumcircle of $\triangle CED$ and line that is parallel to $AB$ and passing through C. Let $G$ be a point of intersection of line $FD$ and side $AB$, and let $H$ be on line $AB$ such that $\angle HDA = \angle GEB$ and $H-A-B$. If $DG=EH$, prove that point of intersection of $AD$ and $BE$ lie on angle bisector of $\angle ACB$.



Proposed by Milos Milosavljevic
2014 Serbia National Math Olympiad
Two players are playing game. Players alternately write down one natural number greater than $1$, but it is not allowed to write linear combination previously written numbers with nonnegative integer coefficients. Player lose a game if he can't write a new number. Does any of players can have wiining strategy, if yes, then which one of them?



Journal "Kvant" / Aleksandar Ilic
2014 Serbia National Math Olympiad
We call natural number $n$ $crazy$ iff there exist natural numbers $a$, $b >1$ such that $n=a^b+b$. Whether there exist $2014$ consecutive natural numbers among which are $2012$ $crazy$ numbers?



Proposed by Milos Milosavljevic
2014 Serbia National Math Olympiad
Regular $n$-gon is divided to triangles using $n-3$ diagonals of which none of them have common points with another inside polygon. How much among this triangles can there be the most not congruent?



Proposed by Dusan Djukic
2014 Serbia National Math Olympiad
In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{  \angle BAC, \angle ABC  \}$



Proposed by Dusan Djukic $IMO \ Shortlist \ 2013$
2015 Serbia National Math Olympiad
Consider circle inscribed quadriateral $ABCD$. Let $M,N,P,Q$ be midpoints of sides $DA,AB,BC,CD$.Let $E$ be the point of intersection of diagonals. Let $k1,k2$ be circles around $EMN$ and $EPQ$ . Let $F$ be point of intersection of $k1$ and $k2$ different from $E$. Prove that $EF$ is perpendicular to $AC$.
2015 Serbia National Math Olympiad
Let $k$ be fixed positive integer .

Let $Fk(n)$ be smallest positive integer bigger than $kn$ such that $Fk(n)*n$ is a perfect square .

Prove that if $Fk(n)=Fk(m)$ than $m=n$.
2015 Serbia National Math Olympiad
We have $2015$ prisinoers.The king gives everyone a hat coloured in one of $5$ colors.Everyone sees all hats expect his own.Now,the King orders them in a line(a prisioner can  see all guys behind and in front of him).The king asks the prisinoers

one by one does he know the color of his hat.If he answers NO,then he is killed.If he answers YES,then answers which color is his hat,if his answers is true,he goes to freedom,if not,he is killed.All the prisinors can hear did he answer YES or NO,but if he answered YES,they don't know what did he answered(he is killed in public).They can think of a strategy before the King comes,but after that they can't comunicate.What is the largest number of prisinors we can guarentee that can survive?
2015 Serbia National Math Olympiad
For integer $a$, $a \neq 0$, $v_2(a)$ is greatest nonnegative integer $k$ such that $2^k | a$. For given $n \in \mathbb{N}$ determine highest possible cardinality of subset $A$ of set $ \{1,2,3,...,2^n \} $ with following property:

For all $x, y \in A$, $x \neq y$, number $v_2(x-y)$ is even.
2015 Serbia National Math Olympiad
Let $x,y,z$ be nonnegative positive integers.

Prove $\frac{x-y}{xy+2y+1}+\frac{y-z}{zy+2z+1}+\frac{z-x}{xz+2x+1}\ge 0$
2015 Serbia National Math Olympiad
In nonnegative set of integers solve the equation:

$$(2^{2015}+1)^x + 2^{2015}=2^y+1$$
2016 Serbia National Math Olympiad
Let $n>1$ be an integer. Prove that there exist $m>n^n $ such that $\frac {n^m-m^n}{m+n} $ is a positive integer.
2016 Serbia National Math Olympiad
Let $n $ be a positive integer. Let $f $ be a function from nonnegative integers to themselves. Let $f (0,i)=f (i,0)=0$, $f (1, 1)=n $, and $ f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}] $ for positive integers $i, j$ such that $i*j>1$. Find the number of pairs $(i,j) $ such that $f (i, j) $ is an odd number.( $[x]$ is the floor function).
2016 Serbia National Math Olympiad
Let $ABC$ be a triangle and $O$ its circumcentre. A line tangent to the circumcircle of the triangle $BOC$ intersects sides $AB$ at $D$ and $AC$ at $E$. Let $A'$ be the image of $A$ under $DE$. Prove that the circumcircle of the triangle $A'DE$ is tangent to the circumcircle of triangle $ABC$.
2016 Serbia National Math Olympiad
Let $ABC $be a triangle, and $I $ the incenter, $M $ midpoint of $ BC $, $ D $ the touch point of incircle and $ BC $. Prove that perpendiculars from $M, D, A $ to $AI, IM, BC $ respectively are concurrent
2016 Serbia National Math Olympiad
There are $2n-1$ twoelement subsets of set $1,2,...,n$. Prove that one can choose $n$ out of these such that their union contains no more than $\frac{2}{3}n+1$ elements.
2016 Serbia National Math Olympiad
Let $a_1, a_2, \dots, a_{2^{2016}}$ be positive integers not bigger than  $2016$. We know that for each $n \leq 2^{2016}$, $a_1a_2 \dots a_{n} +1 $ is a perfect square. Prove that for some $i $ , $a_i=1$.
2017 Serbia National Math Olympiad
Prove that for positive real numbers $a,b,c$ such that $a+b+c=1$,

$$a\sqrt{2b+1}+b\sqrt{2c+1}+c\sqrt{2a+1}\le \sqrt{2-(a^2+b^2+c^2)}.$$
2017 Serbia National Math Olympiad
Let $ABCD$ be a convex and cyclic quadrilateral.

Let $AD\cap BC=\{E\}$, and let $M,N$ be points on $AD,BC$ such that $AM:MD=BN:NC$. Circle around $\triangle EMN$ intersects circle around $ABCD$ at $X,Y$ prove that $AB,CD$ and $XY$ are either parallel or concurrent.
2017 Serbia National Math Olympiad
There are $2n-1$ lamps in a row. In the beginning only the middle one is on (the $n$-th one), and the other ones are off. Allowed move is to take two non-adjacent lamps which are turned off such that all lamps between them are turned on and switch all of their states from on to off and vice versa. What is the maximal number of moves until the process terminates?
2017 Serbia National Math Olympiad
Let $a$ be a positive integer.Suppose that $\forall n$ ,$\exists d$, $d\not =1$, $d\equiv 1\pmod n$ ,$d\mid n^2a-1$.Prove that $a$ is a perfect square.
2017 Serbia National Math Olympiad
Find the maximum number of queens you could put on $2017 \times 2017$ chess table such that each queen attacks at most $1$ other queen.
2017 Serbia National Math Olympiad
Let $k$ be the circumcircle of $\triangle ABC$ and let $k_a$ be A-excircle .Let the two common tangents of $k,k_a$ cut $BC$ in $P,Q$.Prove that $\measuredangle PAB=\measuredangle CAQ$.
2018 Serbia National Math Olympiad
Let $\triangle ABC$ be a triangle with incenter $I$. Points $P$ and $Q$ are chosen on segmets $BI$ and $CI$ such that $2\angle  PAQ=\angle BAC$. If $D$ is the touch point of incircle and side $BC$ prove that $\angle PDQ=90$.
2018 Serbia National Math Olympiad
Let $n>1$ be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by $n$. Prove that the number of consecutive beautiful numbers is less or equal to $1+\lfloor \sqrt{3n} \rfloor$.
2018 Serbia National Math Olympiad
Let $n$ be a positive integer. There are given $n$ lines such that no two are parallel and no three meet at a single point.

a) Prove that there exists a line such that the number of intersection points of these $n$ lines on both of its sides is at least

$$\left \lfloor \frac{(n-1)(n-2)}{10} \right \rfloor.$$Notice that the points on the line are not counted.

b) Find all $n$ for which there exists a configurations where the equality is achieved.
2018 Serbia National Math Olympiad
Prove that there exists a uniqe $P(x)$ polynomial with real coefficients such that

$xy-x-y|(x+y)^{1000}-P(x)-P(y)$ for all real $x,y$.
2018 Serbia National Math Olympiad
Let $a,b>1$ be odd positive integers. A board with $a$ rows and $b$ columns without fields $(2,1),(a-2,b)$ and $(a,b)$ is tiled with $2\times 2$ squares and $2\times 1$ dominoes (that can be rotated). Prove that the number of dominoes is at least $$\frac{3}{2}(a+b)-6.$$
2018 Serbia National Math Olympiad
For each positive integer $k$, let $n_k$ be the smallest positive integer such that there exists a finite set $A$ of integers satisfy the following properties:



For every $a\in A$, there exists $x,y\in A$ (not necessary distinct) that

$$n_k\mid a-x-y$$

There's no subset $B$ of $A$ that $|B|\leq k$ and $$n_k\mid \sum_{b\in B}{b}.$$



Show that for all positive integers $k\geq 3$, we've $$n_k<\Big( \frac{13}{8}\Big)^{k+2}.$$
2019 Serbia National Math Olympiad
Find all positive integers $n, n>1$ for wich holds :

If $a_1, a_2 ,\dots ,a_k$ are all numbers less than $n$ and relatively prime to $n$ , and holds $a_1<a_2<\dots <a_k $, then none of sums $a_i+a_{i+1}$ for $i=1,2,3,\dots k-1 $ are divisible by $3$.
2019 Serbia National Math Olympiad
For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is invested on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ .

A sequence is  normed if all its members are not greater than $1$ . For a given natural $n$ , prove :



a)Every normed sequence of length $2n+1$ is invested in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$.

b) there exists normed sequence of length $4n+3$ wich is not invested on $\left[ 0, 2-\frac{1}{2^n} \right ]$.
2019 Serbia National Math Olympiad
Let $k$ be the circle inscribed in convex quadrilateral $ABCD$. Lines $AD$ and $BC$  meet at $P$ ,and circumcircles of $\triangle PAB$ and $\triangle PCD$ meet in $X$ . Prove that tangents from $X$ to $k$ form equal angles with lines $AX$ and $CX$ .
2019 Serbia National Math Olympiad
For a  $\triangle ABC$ , let $A_1$ be the symmetric point of the intersection of angle bisector of $\angle BAC$ and $BC$ , where center of the symmetry is the midpoint of side $BC$, In the same way we define $B_1 $ ( on $AC$ ) and $C_1$ (on $AB$). Intersection of circumcircle of $\triangle A_1B_1C_1$ and line $AB$ is the set $\{Z,C_1 \}$, with $BC$ is the set $\{X,A_1\}$ and with $CA$ is the set $\{Y,B_1\}$. If the perpendicular lines from $X,Y,Z$ on $BC,CA$ and $ AB$ , respectively are concurrent , prove that $\triangle ABC$ is isosceles.
2019 Serbia National Math Olympiad
In the spherical shaped planet $X$ there are $2n$ gas stations. Every station is paired with one other station ,

and every two paired stations are diametrically opposite points on the planet.

Each station has a given amount of gas. It is known that : if a car with empty (large enough) tank starting

from any station it is always to reach the paired station with the initial station (it can get extra gas during the journey).

Find all naturals $n$ such that for any placement of $2n$ stations for wich holds the above condotions, holds:

there always a gas station wich the car can start with empty tank and go to all other  stations on the planet.(Consider that the car consumes a constant amount of gas per unit length.)
2019 Serbia National Math Olympiad
Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations :

$$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$and

$$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$Prove that  :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$
2020 Serbia National Math Olympiad
Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$.
2020 Serbia National Math Olympiad
We are given a polyhedron with at least $5$ vertices, such that exactly $3$ edges meet in each of the vertices. Prove that we can assign a rational number to every vertex of the given polyhedron such that the following conditions are met:



$(i)$ At least one of the numbers assigned to the vertices is equal to $2020$.



$(ii)$ For every polygonal face, the product of the numbers assigned to the vertices of that face is equal to $1$.
2020 Serbia National Math Olympiad
We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.
2020 Serbia National Math Olympiad
In a trapezoid $ABCD$ such that the internal angles are not equal to $90^{\circ}$, the diagonals $AC$ and $BD$ intersect at the point $E$. Let $P$ and $Q$ be the feet of the altitudes from $A$ and $B$ to the sides $BC$ and $AD$ respectively. Circumscribed circles of the triangles $CEQ$ and $DEP$ intersect at the point $F\neq E$. Prove that the lines $AP$, $BQ$ and $EF$ are either parallel to each other, or they meet at exactly one point.
2020 Serbia National Math Olympiad
For a natural number $n$,  with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions:



$(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$.

$(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$.



Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.
2020 Serbia National Math Olympiad
We are given a natural number $k$. Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute $n$ coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves:





$(i)$ We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other.

$(ii)$ We choose a field with at least $2$ coins on it, and we transfer one coin from the chosen field to the $k-\mathrm{th}$ field on the left , and one coin from the chosen field to the $k-\mathrm{th}$ field on the right.



$\mathbf{(a)}$ If $n\leq k+1$, prove that we can play only finitely many moves.

$\mathbf{(b)}$ For which values of $k$ we can choose a natural number $n$ and distribute $n$ coins on the given board such that we can play infinitely many moves.
2021 Serbia National Math Olympiad
Let $a>1$ and $c$ be natural numbers and let $b\neq 0$ be an integer. Prove that there exists a natural number $n$ such that the number $a^n+b$ has a divisor of the form $cx+1$, $x\in\mathbb{N}$.
2021 Serbia National Math Olympiad
In the country of Graphia there are $100$ towns, each numbered from $1$ to $100$. Some pairs of towns may be connected by a (direct) road and we call such pairs of towns  adjacent. No two roads connect the same pair of towns.



Peter, a foreign tourist, plans to visit Graphia $100$ times. For each $i$, $i=1,2,\dots, 100$, Peter starts his $i$-th trip by arriving in the town numbered $i$ and then each following day Peter travels from the town he is currently in to an adjacent town with the lowest assigned number, assuming such that a town exists and that he hasn't visited it already on the $i$-th trip. Otherwise, Peter deems his $i$-th trip to be complete and returns home.



It turns out that after all $100$ trips, Peter has visited each town in Graphia the same number of times. Find the largest possible number of roads in Graphia.
2021 Serbia National Math Olympiad
In a triangle $ABC$, let $AB$ be the shortest side. Points $X$ and $Y$ are given on the circumcircle of $\triangle ABC$ such that $CX=AX+BX$ and $CY=AY+BY$. Prove that $\measuredangle XCY<60^{o}$.
2021 Serbia National Math Olympiad
A convex quadrilateral $ABCD$ will be called rude if there exists a convex quadrilateral $PQRS$ whose points are all in the interior or on the sides of quadrilateral $ABCD$ such that the sum of diagonals of $PQRS$ is larger than the sum of diagonals of $ABCD$.



Let $r>0$ be a real number. Let us assume that a convex quadrilateral $ABCD$ is not rude, but every quadrilateral $A'BCD$ such that $A'\neq A$ and $A'A\leq r$ is rude. Find all possible values of the largest angle of $ABCD$.
2021 Serbia National Math Olympiad
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for every $x,y\in\mathbb{R}$ the following equality holds: $$f(xf(y)+x^2+y)=f(x)f(y)+xf(x)+f(y).$$
2021 Serbia National Math Olympiad
A finite sequence of natural numbers $a_1, a_2, \dots, a_n$ is given. A sub-sequence $a_{k+1}, a_{k+2}, \dots, a_l$ will be called a repetition if there exists a natural number $p\leq \frac{l-k}2$ such that $a_i=a_{i+p}$ for $k+1\leq i\leq l-p$, but $a_i\neq a_{i+p}$ for $i=k$ (if $k>0$) and $i=l-p+1$ (if $l<n$).



Show that the sequence contains less than $n$ repetitions.