Navvye/Olympiad-Problems-CSV huggingface dataset:
2003 Bulgaria National Olympiad
Let $H$ be an arbitrary point on the altitude $CP$ of the acute triangle $ABC$. The lines $AH$ and $BH$ intersect $BC$ and $AC$ in $M$ and $N$, respectively.



(a) Prove that

$\angle NPC =\angle MPC$.

(b) Let $O$ be the common point of $MN$ and $CP$. An arbitrary line through $O$ meets the sides of quadrilateral $CNHM$ in $D$ and $E$. Prove that $\angle EPC =\angle DPC$.
2003 Bulgaria National Olympiad
Given the sequence $\{y_n\}_{n=1}^{\infty}$ defined by $y_1=y_2=1$ and

$$y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1$$

find all integers $k$ such that every term of the sequence is a perfect square.
2003 Bulgaria National Olympiad
A set $A$ of positive integers is called uniform if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer $n>1$ such that there exist a uniform set $A$ with $n$ elements.
2003 Bulgaria National Olympiad
Let $a,b,c$ be rational numbers such that $a+b+c$ and $a^2+b^2+c^2$ are equal integers. Prove that the number $abc$ can be written as the ratio of a perfect cube and  a perfect square which are relatively prime.
2003 Bulgaria National Olympiad
Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.
2004 Bulgaria National Olympiad
Let $ I$ be the incenter of triangle $ ABC$, and let $ A_1$, $ B_1$, $ C_1$ be arbitrary points on the segments $ (AI)$, $ (BI)$, $ (CI)$, respectively. The perpendicular bisectors of $ AA_1$, $ BB_1$, $ CC_1$ intersect each other at $ A_2$, $ B_2$, and $ C_2$. Prove that the circumcenter of the triangle $ A_2B_2C_2$ coincides with the circumcenter of the triangle $ ABC$ if and only if $ I$ is the orthocenter of triangle $ A_1B_1C_1$.
2004 Bulgaria National Olympiad
For any positive integer $n$ the sum $\displaystyle 1+\frac 12+ \cdots + \frac 1n$ is written in the form $\displaystyle \frac{P(n)}{Q(n)}$, where $P(n)$ and $Q(n)$ are relatively prime.



a) Prove that $P(67)$ is not divisible by 3;



b) Find all possible $n$, for which $P(n)$ is divisible by 3.
2004 Bulgaria National Olympiad
A group consist of n tourists. Among every 3 of them there are 2 which are not familiar. For every partition of the tourists in 2 buses you can find 2 tourists that are in the same bus and are familiar with each other. Prove that is a tourist familiar to at most $\displaystyle \frac 2{5}n$ tourists.
2004 Bulgaria National Olympiad
In a word formed with the letters $a,b$ we can change some blocks: $aba$ in $b$ and back, $bba$ in $a$ and backwards. If the initial word is $aaa\ldots ab$ where $a$ appears 2003 times can we reach the word $baaa\ldots a$, where $a$ appears 2003 times.
2004 Bulgaria National Olympiad
Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$.
2004 Bulgaria National Olympiad
Let $ p$ be a prime number and let $ 0\leq a_{1}< a_{2}<\cdots < a_{m}< p$ and $ 0\leq b_{1}< b_{2}<\cdots < b_{n}< p$ be arbitrary integers. Let $ k$ be the number of distinct residues modulo $ p$ that $ a_{i}+b_{j}$ give when $ i$ runs from 1 to $ m$, and $ j$ from 1 to $ n$. Prove that



a) if $ m+n > p$ then $ k = p$;



b) if $ m+n\leq p$ then $ k\geq m+n-1$.
2005 Bulgaria National Olympiad
Determine all triples $\left( x,y,z\right)$ of positive integers for which the number $$ \sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}} $$ is an integer .
2005 Bulgaria National Olympiad
Consider two circles $k_{1},k_{2}$ touching externally at point $T$. a line touches $k_{2}$ at point $X$ and intersects $k_{1}$ at points $A$ and $B$. Let $S$ be the second intersection point of $k_{1}$ with the line $XT$ . On the arc $\widehat{TS}$ not containing $A$ and $B$ is chosen a point $C$ . Let $\ CY$ be the tangent line to $k_{2}$ with $Y\in k_{2}$ , such that the segment $CY$ does not intersect the segment $ST$ . If $I=XY\cap SC$ . Prove that :



(a) the points $C,T,Y,I$ are concyclic.



(b) $I$ is the excenter of triangle $ABC$ with respect to the side $BC$.
2005 Bulgaria National Olympiad
Let $M=(0,1)\cap \mathbb Q$. Determine, with proof, whether there exists a subset $A\subset M$ with the property that every number in $M$ can be uniquely written as the sum of finitely many distinct elements of $A$.
2005 Bulgaria National Olympiad
Let $ABC$ be a triangle with $AC\neq BC$, and let $A^{\prime }B^{\prime }C$ be a triangle obtained from $ABC$ after some rotation centered at $C$. Let $M,E,F$ be the midpoints of the segments $BA^{\prime },AC$ and $CB^{\prime }$  respectively. If $EM=FM$, find $\widehat{EMF}$.
2005 Bulgaria National Olympiad
For positive integers $t,a,b,$a $(t,a,b)$-game is a two player game defined by the following rules. Initially, the number $t$ is written on a blackboard. At his first move, the 1st player replaces $t$ with either $t-a$ or $t-b$. Then, the 2nd player subtracts either $a$ or $b$ from this number, and writes the result on the blackboard, erasing the old number. After this, the first player once again erases either $a$ or $b$ from the number written on the blackboard, and so on. The player who first reaches a negative number loses the game. Prove that there exist infinitely many values of $t$ for which the first player has a winning strategy for all pairs $(a,b)$ with $a+b=2005$.
2005 Bulgaria National Olympiad
Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$.

Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide.
2006 Bulgaria National Olympiad
Consider the set $A=\{1,2,3\ldots ,2^n\}, n\ge 2$. Find the number of subsets $B$ of $A$ such that for any two elements of $A$ whose sum is a power of $2$ exactly one of them is in $B$.



Aleksandar Ivanov
2006 Bulgaria National Olympiad
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$

$$f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}$$

a) Prove that $f(2x)=4f(x)$ for all $x>0$;

b) Find all such functions.



Nikolai Nikolov, Oleg Mushkarov
2006 Bulgaria National Olympiad
The natural numbers are written in sequence, in increasing order, and by this we get an infinite sequence of digits. Find the least natural $k$, for which among the first $k$ digits of this sequence, any two nonzero digits have been written a different number of times.



Aleksandar Ivanov, Emil Kolev
2006 Bulgaria National Olympiad
Let $p$ be a prime such that $p^2$ divides $2^{p-1}-1$. Prove that for all positive integers $n$ the number $\left(p-1\right)\left(p!+2^n\right)$ has at least $3$ different prime divisors.



Aleksandar Ivanov
2006 Bulgaria National Olympiad
The triangle $ABC$ is such that $\angle BAC=30^{\circ},\angle ABC=45^{\circ}$. Prove that if $X$ lies on the ray $AC$, $Y$ lies on the ray $BC$ and $OX=BY$, where $O$ is the circumcentre of triangle $ABC$, then $S_{XY}$ passes through a fixed point.



Emil Kolev
2006 Bulgaria National Olympiad
Consider a point $O$ in the plane. Find all sets $S$ of at least two points in the plane such that if $A\in S$ ad $A\neq O$, then the circle with diameter $OA$ is in $S$.



Nikolai Nikolov, Slavomir Dinev
2007 Bulgaria National Olympiad
The quadrilateral $ABCD$, where $\angle BAD+\angle ADC>\pi$, is inscribed a circle with centre $I$. A line through $I$ intersects $AB$ and $CD$ in points $X$ and $Y$ respectively such that $IX=IY$. Prove that $AX\cdot DY=BX\cdot CY$.
2007 Bulgaria National Olympiad
Find the greatest positive integer $n$ such that we can choose $2007$ different positive integers from $[2\cdot 10^{n-1},10^{n})$ such that for each two $1\leq i<j\leq n$ there exists a positive integer $\overline{a_{1}a_{2}\ldots a_{n}}$ from the chosen integers for which $a_{j}\geq a_{i}+2$.



A. Ivanov, E. Kolev
2007 Bulgaria National Olympiad
Find the least positive integer $n$ such that $\cos\frac{\pi}{n}$ cannot be written in the form $p+\sqrt{q}+\sqrt[3]{r}$ with $p,q,r\in\mathbb{Q}$.



O. Mushkarov, N. Nikolov



Click to reveal hidden textNo-one in the competition scored more than 2 points
2007 Bulgaria National Olympiad
Let $k>1$ be a given positive integer. A set $S$ of positive integers is called good if we can colour the set of positive integers in $k$ colours such that each integer of $S$ cannot be represented as sum of two positive integers of the same colour. Find the greatest $t$ such that the set $S=\{a+1,a+2,\ldots ,a+t\}$ is good for all positive integers $a$.



A. Ivanov, E. Kolev
2007 Bulgaria National Olympiad
Find the least real number $m$ such that with all $5$ equilaterial triangles with sum of areas $m$ we can cover an equilaterial triangle with side 1.



O. Mushkarov, N. Nikolov
2007 Bulgaria National Olympiad
Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.
2008 Bulgaria National Olympiad
Let $ ABC$ be an acute triangle and $ CL$ be the angle bisector of $ \angle ACB$. The point $ P$ lies on  the segment $CL$ such that $ \angle APB=\pi-\frac{_1}{^2}\angle ACB$. Let $ k_1$ and $ k_2$ be the circumcircles of the triangles $ APC$ and $ BPC$. $ BP\cap k_1=Q, AP\cap k_2=R$. The tangents to $ k_1$ at $ Q$ and $ k_2$ at $ B$ intersect at $ S$ and the tangents to $ k_1$ at $ A$ and $ k_2$ at $ R$ intersect at $ T$. Prove that $ AS=BT.$
2008 Bulgaria National Olympiad
Is it possible to find $2008$ infinite arithmetical progressions such that there exist finitely many positive integers not in any of these progressions, no two progressions intersect and each progression contains a prime number bigger than $2008$?
2008 Bulgaria National Olympiad
Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied :

$$\left|\sum_{i=1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}$$

for all $ k\in\mathbb{N}$. Prove that $ b_1=b_2=\ldots =b_n=0$.
2008 Bulgaria National Olympiad
Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 + 279m + 5^n$ is $ k$-th power of some natural number.
2008 Bulgaria National Olympiad
Let $n$ be a fixed natural number. Find all natural numbers $ m$ for which

$$\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m$$

is satisfied for every two positive numbers $ a$ and $ b$ with sum equal to $2$.
2008 Bulgaria National Olympiad
Let $M$ be the set of the integer numbers from the range $[-n, n]$. The subset $P$ of $M$ is called a base subset if every number from $M$ can be expressed as a sum of some different numbers from $P$. Find the smallest natural number $k$ such that every $k$ numbers that belongs to $M$ form a base subset.
2009 Bulgaria National Olympiad
The natural numbers $a$ and $b$ satisfy the inequalities $a > b > 1$ .  It is also known that the equation

$\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}$ has at least two solutions in natural numbers, when $x > 1$ and $y > 1$.

Prove that the numbers $a$ and $b$ are coprime (their greatest common divisor is $1$).
2009 Bulgaria National Olympiad
In the triangle $ABC$ its incircle with center $I$ touches its sides $BC, CA$ and $AB$ in the points $A_1, B_1, C_1$ respectively. Through $I$ is drawn a line $\ell$. The points $A', B'$ and $C'$ are reflections of $A_1, B_1, C_1$ with respect to the line $\ell$. Prove that the lines $AA', BB'$ and $CC'$ intersects at a common point.
2009 Bulgaria National Olympiad
Through the points with integer coordinates in the right-angled coordinate system $Oxyz$ are constructed planes, parallel to the coordinate planes and in this way the space is divided to unit cubes. Find all triples ($a, b, c$) consisting of natural numbers ($a \le  b \le c$) for which the cubes formed can be coloured in $abc$ colors in such a way that every palellepiped with  dimensions $a \times  b \times c$, having vertices with integer coordinates and sides parallel to the coordinate axis doesn't contain unit cubes in the same color.
2009 Bulgaria National Olympiad
Let $ n\ge 3$ be a natural number. Find all nonconstant polynomials with real coeficcietns $ f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$, for which

$$ f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), \quad  1\le k\le n,$$

for every real $ x$ (with $ f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$ and $ f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$).
2009 Bulgaria National Olympiad
We divide a convex $2009$-gon in triangles using non-intersecting diagonals. One of these diagonals is colored green. It is allowed the following operation: for two triangles $ABC$ and $BCD$ from the dividing/separating with a common side $BC$ if the replaced diagonal was green it loses its color and the replacing diagonal becomes green colored. Prove that if we choose any diagonal in advance it can be colored in green after applying the operation described finite number of times.
2009 Bulgaria National Olympiad
Prove that if $ a_{1},a_{2},\ldots,a_{n}$, $ b_{1},b_{2},\ldots,b_{n}$ are arbitrary taken real numbers and $ c_{1},c_{2},\ldots,c_{n}$

are positive real numbers, than

$ \left(\sum_{i,j = 1}^{n}\frac {a_{i}a_{j}}{c_{i} + c_{j}}\right)\left(\sum_{i,j = 1}^{n}\frac {b_{i}b_{j}}{c_{i} + c_{j}}\right)\ge \left(\sum_{i,j = 1}^{n}\frac {a_{i}b_{j}}{c_{i} + c_{j}}\right)^{2}$.
2010 Bulgaria National Olympiad
A table $2 \times 2010$ is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal $2 \times 1$ domino that covers exactly two unit table cells. Then Peter puts vertical $1 \times 2$ domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.
2010 Bulgaria National Olympiad
Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.
2010 Bulgaria National Olympiad
Let $a_0, a_1, \ldots, a_9$ and $b_1 , b_2, \ldots,b_9$ be positive integers such that $a_9<b_9$ and $a_k \neq b_k, 1 \leq k \leq 8.$ In a cash dispenser/automated teller machine/ATM there are $n\geq a_9$ levs (Bulgarian national currency) and for each $1 \leq i \leq 9$ we can take $a_i$ levs from the ATM  (if in the bank there are at least $a_i$ levs).  Immediately after that action the bank puts $b_i$ levs in the ATM or we take $a_0$ levs. If we take $a_0$ levs from the ATM the bank doesn’t put any money in the ATM. Find all possible positive integer values of $n$ such that after finite number of takings money from the ATM there will be no money in it.
2010 Bulgaria National Olympiad
Does there exist a number $n=\overline{a_1a_2a_3a_4a_5a_6}$ such that $\overline{a_1a_2a_3}+4 = \overline{a_4a_5a_6}$ (all bases are $10$) and $n=a^k$ for some positive integers $a,k$ with $k \geq 3 \ ?$
2010 Bulgaria National Olympiad
Let $f: \mathbb N \to \mathbb N$ be a function such that $f(1)=1$ and

$$f(n)=n - f(f(n-1)), \quad \forall n \geq 2.$$Prove that $f(n+f(n))=n $ for each positive integer $n.$
2010 Bulgaria National Olympiad
Let $k$ be the circumference of the triangle $ABC.$ The point $D$ is an arbitrary point on the segment $AB.$ Let $I$ and $J$ be the centers of the circles which are tangent to the side $AB,$ the segment $CD$ and the circle $k.$ We know that the points $A, B, I$ and $J$ are concyclic. The excircle of the triangle $ABC$  is tangent to the side $AB$ in the point $M.$ Prove that $M \equiv D.$
2011 Bulgaria National Olympiad
Prove whether or not there exist natural numbers $n,k$ where $1\le k\le n-2$ such that

$$\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4 $$
2011 Bulgaria National Olympiad
Let $f_1(x)$ be a polynomial of degree $2$ with the leading coefficient positive and $f_{n+1}(x) =f_1(f_n(x))$ for $n\ge 1.$ Prove that if the equation $f_2(x)=0$ has four different non-positive real roots, then for arbitrary $n$ then $f_n(x)$ has $2^n$ different real roots.
2011 Bulgaria National Olympiad
Triangle $ABC$ and a function $f:\mathbb{R}^+\to\mathbb{R}$ have the following property: for every line segment $DE$ from the interior of the triangle with midpoint $M$, the inequality $f(d(D))+f(d(E))\le 2f(d(M))$, where $d(X)$ is the distance from point $X$ to the nearest side of the triangle ($X$ is in the interior of $\triangle ABC$). Prove that for each line segment $PQ$ and each point interior point $N$ the inequality $|QN|f(d(P))+|PN|f(d(Q))\le |PQ|f(d(N))$ holds.
2011 Bulgaria National Olympiad
Point $O$ is inside $\triangle ABC$. The feet of perpendicular from $O$ to $BC,CA,AB$ are $D,E,F$. Perpendiculars from $A$ and $B$ respectively to $EF$ and $FD$ meet at $P$. Let $H$ be the foot of perpendicular from $P$ to $AB$. Prove that $D,E,F,H$ are concyclic.
2011 Bulgaria National Olympiad
For each natural number $a$ we denote $\tau (a)$ and $\phi (a)$ the number of natural numbers dividing $a$ and the number of natural numbers less than $a$ that are relatively prime to $a$. Find all natural numbers $n$ for which $n$ has exactly two different prime divisors and $n$ satisfies $\tau (\phi (n))=\phi (\tau (n))$.
2011 Bulgaria National Olympiad
In the interior of the convex 2011-gon are $2011$ points, such that no three among the given $4022$ points (the interior points and the vertices) are collinear. The points are coloured one of two different colours and a colouring is called "good" if some of the points can be joined  in such a way that the following conditions are satisfied:

1) Each segment joins two points of the same colour.

2) None of the line segments intersect.

3) For any two points of the same colour there exists a path of segments connecting them.

Find the number of "good" colourings.
2012 Bulgaria National Olympiad
The sequence $a_1,a_2,a_3\ldots $, consisting of natural numbers, is defined by the rule:

$$a_{n+1}=a_{n}+2t(n)$$

for every natural number $n$, where $t(n)$ is the number of the different divisors of $n$ (including $1$ and $n$). Is it possible that two consecutive members of the sequence are squares of natural numbers?
2012 Bulgaria National Olympiad
Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled:

1) For every prime number $p$ and every natural number $n$, the numbers $p^n,p^{n+1}$ and $p^{n+2}$ do not have the same colour.

2) There does not exist an infinite geometric sequence of natural numbers of the same colour.
2012 Bulgaria National Olympiad
We are given a real number $a$, not equal to $0$ or $1$. Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation:

$$*x^4+*x^3+*x^2+*x^1+*=0$$

with a number of the type $a^n$, where $n$ is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of $a$) who has a winning strategy
2012 Bulgaria National Olympiad
Let $n$ be an even natural number and let $A$ be the set of all non-zero sequences of length $n$, consisting of numbers $0$ and $1$ (length $n$ binary sequences, except the zero sequence $(0,0,\ldots,0)$). Prove that $A$ can be partitioned into groups of three elements, so that for every triad $\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}$, and for every $i = 1, 2,\ldots,n$, exactly zero or two of the numbers $a_i, b_i, c_i$ are equal to $1$.
2012 Bulgaria National Olympiad
Let $Q(x)$ be a quadratic trinomial. Given that the function $P(x)=x^{2}Q(x)$ is increasing in the interval $(0,\infty )$, prove that:

$$P(x) + P(y) + P(z) > 0$$

for all real numbers $x,y,z$ such that $x+y+z>0$ and $xyz>0$.
2012 Bulgaria National Olympiad
We are given an acute-angled triangle $ABC$ and a random point $X$ in its interior, different from the centre of the circumcircle $k$ of the triangle. The lines $AX,BX$ and $CX$ intersect $k$ for a second time in the points $A_1,B_1$ and $C_1$ respectively. Let $A_2,B_2$ and $C_2$ be the points that are symmetric of  $A_1,B_1$ and $C_1$ in respect to $BC,AC$ and $AB$ respectively. Prove that the circumcircle of the triangle $A_2,B_2$ and $C_2$ passes through a constant point that does not depend on the choice of $X$.
2013 Bulgaria National Olympiad
Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square.



Proposed by P. Boyvalenkov
2013 Bulgaria National Olympiad
Find all $f : \mathbb{R}\to \mathbb{R}$ , bounded in $(0,1)$ and satisfying:



$x^2 f(x) - y^2 f(y) = (x^2-y^2) f(x+y) -xy f(x-y)$



for all $x,y \in \mathbb{R}$



Proposed by Nikolay Nikolov
2013 Bulgaria National Olympiad
The integer lattice in the plane is colored with 3 colors. Find the least positive real $S$ with the property: for any such coloring it is possible to find a monochromatic lattice points $A,B,C$ with $S_{\triangle ABC}=S$.



Proposed by Nikolay Beluhov



EDIT: It was the problem 3 (not 2), corrected the source title.
2013 Bulgaria National Olympiad
Suppose $\alpha,\beta,\gamma \in [0.\pi/2)$ and $\tan \alpha + \tan\beta + \tan \gamma \leq 3$.

Prove that:

$$\cos 2\alpha + \cos 2\beta + \cos 2\gamma \ge 0$$



Proposed by Nikolay Nikolov
2013 Bulgaria National Olympiad
Consider acute $\triangle ABC$ with altitudes $AA_1, BB_1$ and $CC_1$ ($A_1 \in BC,B_1 \in  AC,C_1 \in  AB$). A point $C' $ on the extension of $B_1A_1$ beyond $A_1$ is such that $A_1C' = B_1C_1$. Analogously, a point $B'$ on the extension of A$_1C_1$ beyond $C_1$ is such that $C_1B'  = A_1B_1$ and a point $A' $ on the extension of $C_1B_1$ beyond $B_1$ is such that $B_1A'  = C_1A_1$. Denote by $A'', B'', C''$ the symmetric points of $A' , B' , C'$  with respect to $BC, CA$ and $AB$ respectively. Prove that if $R, R'$  and R'' are circumradiii of $\triangle ABC, \triangle A'B'C'$ and $\triangle A''B''C''$, then $R, R'$  and $R'' $ are sidelengths of a triangle with area equals one half of the area of $\triangle ABC$.
2013 Bulgaria National Olympiad
Given $m\in\mathbb{N}$ and a prime number $p$, $p>m$, let

$$M=\{n\in\mathbb{N}\mid m^2+n^2+p^2-2mn-2mp-2np \,\,\, \text{is a perfect square}  \} $$

Prove that $|M|$ does not depend on $p$.





Proposed by Aleksandar Ivanov
2014 Bulgaria National Olympiad
Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies.



Proposed by T. Vitanov, E. Kolev
2014 Bulgaria National Olympiad
Every cell of a $m \times n$ chess board, $m\ge 2,n\ge 2$, is colored with one of four possible colors, e.g white, green, red, blue. We call such coloring good if the four cells of any $2\times 2$ square of the chessboard are colored with pairwise different colors. Determine the number of all good colorings of the chess board.



Proposed by N. Beluhov
2014 Bulgaria National Olympiad
A real number $f(X)\neq 0$ is assigned to each point $X$ in the space.

It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have :

$$ f(O)=f(A)f(B)f(C)f(D). $$

Prove that $f(X)=1$ for all points $X$.



Proposed by Aleksandar Ivanov
2014 Bulgaria National Olympiad
Find all pairs of prime numbers $p\,,q$ for which:

$$p^2 \mid q^3 + 1 \,\,\, \text{and} \,\,\, q^2 \mid p^6-1$$



Proposed by P. Boyvalenkov
2014 Bulgaria National Olympiad
Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property:

$$f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+$$



Proposed by Nikolay Nikolov