Navvye/Olympiad-Problems-CSV huggingface dataset:
1987 Bulgaria National Olympiad
Let $E$ be a point on the median $AD$ of a triangle $ABC$, and $F$ be the projection of $E$ onto $BC$. From a point $M$ on $EF$ the perpendiculars $MN$ to $AC$ and $MP$ to $AB$ are drawn. Prove that if the points $N,E,P$ lie on a line, then $M$ lies on the bisector of $\angle BAC$.
1987 Bulgaria National Olympiad
Let $\Delta$ be the set of all triangles inscribed in a given circle, with angles whose measures are integer numbers of degrees different than $45^\circ,90^\circ$ and $135^\circ$. For each triangle $T\in\Delta$, $f(T)$ denotes the triangle with vertices at the second intersection points of the altitudes of $T$ with the circle.



(a) Prove that there exists a natural number $n$ such that for every triangle $T\in\Delta$, among the triangles $T,f(T),\ldots,f^n(T)$ (where $f^0(T)=T$ and $f^k(T)=f(f^{k-1}(T))$) at least two are equal.

(b) Find the smallest $n$ with the property from (a).
1988 Bulgaria National Olympiad
Find all real parameters $q$ for which there is a $p\in[0,1]$ such that the equation

$$x^4+2px^3+(2p^2-p)x^2+(p-1)p^2x+q=0$$has four real roots.
1988 Bulgaria National Olympiad
Let $n$ and $k$ be natural numbers and $p$ a prime number. Prove that if $k$ is the exact exponent of $p$ in $2^{2^n}+1$ (i.e. $p^k$ divides $2^{2^n}+1$, but $p^{k+1}$ does not), then $k$ is also the exact exponent of $p$ in $2^{p-1}-1$.
1988 Bulgaria National Olympiad
Let $M$ be an arbitrary interior point of a tetrahedron $ABCD$, and let $S_A,S_B,S_C,S_D$ be the areas of the faces $BCD,ACD,ABD,ABC$, respectively. Prove that

$$S_A\cdot MA+S_B\cdot MB+S_C\cdot MC+S_D\cdot MD\ge9V,$$where $V$ is the volume of $ABCD$. When does equality hold?
1988 Bulgaria National Olympiad
Let $A,B,C$ be non-collinear points. For each point $D$ of the ray $AC$, we denote by $E$ and $F$ the points of tangency of the incircle of $\triangle ABD$ with $AB$ and $AD$, respectively. Prove that, as point $D$ moves along the ray $AC$, the line $EF$ passes through a fixed point.
1988 Bulgaria National Olympiad
The points of space are painted in two colors. Prove that there is a tetrahedron such that all its vertices and its centroid are of the same color.
1988 Bulgaria National Olympiad
Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.
1989 Bulgaria National Olympiad
In triangle $ABC$, point $O$ is the center of the excircle touching the side $BC$, while the other two excircles touch the sides $AB$ and $AC$ at points $M$ and $N$ respectively. A line through $O$ perpendicular to $MN$ intersects the line $BC$ at $P$. Determine the ratio $AB/AC$, given that the ratio of the area of $\triangle ABC$ to the area of $\triangle MNP$ is $2R/r$, where $R$ is the circumradius and $r$ the inradius of $\triangle ABC$.
1989 Bulgaria National Olympiad
Prove that the sequence $(a_n)$, where

$$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.
1989 Bulgaria National Olympiad
Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that:



(a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$;

(b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.
1989 Bulgaria National Olympiad
At each of the given $n$ points on a circle, either $+1$ or $-1$ is written. The following operation is performed: between any two consecutive numbers on the circle their product is written, and the initial $n$ numbers are deleted. Suppose that, for any initial arrangement of $+1$ and $-1$ on the circle, after finitely many operations all the numbers on the circle will be equal to $+1$. Prove that $n$ is a power of two.
1989 Bulgaria National Olympiad
Prove that the perpendiculars, drawn from the midpoints of the edges of the base of a given tetrahedron to the opposite lateral edges, have a common point if and only if the circumcenter of the tetrahedron, the centroid of the base, and the top vertex of the tetrahedron are collinear.
1989 Bulgaria National Olympiad
Let $x,y,z$ be pairwise coprime positive integers and $p\ge5$ and $q$ be prime numbers which satisfy the following conditions:



(i) $6p$ does not divide $q-1$;

(ii) $q$ divides $x^2+xy+y^2$;

(iii) $q$ does not divide $x+y-z$.



Prove that $x^p+y^p\ne z^p$.
1990 Bulgaria National Olympiad
Consider the number obtained by writing the numbers $1,2,\ldots,1990$ one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last remaining digit?
1990 Bulgaria National Olympiad
Let be given a real number $\alpha\ne0$. Show that there is a unique point $P$ in the coordinate plane, such that for every line through $P$ which intersects the parabola $y=\alpha x^2$ in two distinct points $A$ and $B$, segments $OA$ and $OB$ are perpendicular (where $O$ is the origin).
1990 Bulgaria National Olympiad
Let $n=p_1p_2\cdots p_s$, where $p_1,\ldots,p_s$ are distinct odd prime numbers.

(a) Prove that the expression

$$F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},$$where the product goes over all subsets $\{p_{i_1},\ldots,p_{i_k}\}$ or $\{p_1,\ldots,p_s\}$ (including itself and the empty set), can be written as a polynomial in $x$ with integer coefficients.

(b) Prove that if $p$ is a prime divisor of $F_n(2)$, then either $p\mid n$ or $n\mid p-1$.
1990 Bulgaria National Olympiad
Suppose $M$ is an infinite set of natural numbers such that, whenever the sum of two natural numbers is in $M$, one of these two numbers is in $M$ as well. Prove that the elements of any finite set of natural numbers not belonging to $M$ have a common divisor greater than $1$.
1990 Bulgaria National Olympiad
Given a circular arc, find a triangle of the smallest possible area which covers the arc so that the endpoints of the arc lie on the same side of the triangle.
1990 Bulgaria National Olympiad
The base $ABC$ of a tetrahedron $MABC$ is an equilateral triangle, and the lateral edges $MA,MB,MC$ are sides of a triangle of the area $S$. If $R$ is the circumradius and $V$ the volume of the tetrahedron, prove that $RS\ge2V$. When does equality hold?
1991 Bulgaria National Olympiad
Let $M$ be a point on the altitude $CD$ of an acute-angled triangle $ABC$, and $K$ and $L$ the orthogonal projections of $M$ on $AC$ and $BC$. Suppose that the incenter and circumcenter of the triangle lie on the segment $KL$.



(a) Prove that $CD=R+r$, where $R$ and $r$ are the circumradius and inradius, respectively.

(b) Find the minimum value of the ratio $CM:CD$.
1991 Bulgaria National Olympiad
Let $K$ be a cube with edge $n$, where $n>2$ is an even integer. Cube $K$ is divided into $n^3$ unit cubes. We call any set of $n^2$ unit cubes lying on the same horizontal or vertical level a layer. We dispose of $\frac{n^3}4$ colors, in each of which we paint exactly $4$ unit cubes. Prove that we can always select $n$ unit cubes of distinct colors, no two of which lie on the same layer.
1991 Bulgaria National Olympiad
Prove that for every prime number $p\ge5$,



(a) $p^3$ divides $\binom{2p}p-2$;

(b) $p^3$ divides $\binom{kp}p-k$ for every natural number $k$.
1991 Bulgaria National Olympiad
Let $f(x)$ be a polynomial of degree $n$ with real coefficients, having $n$ (not necessarily distinct) real roots. Prove that for all real $x$,

$$f(x)f''(x)\le f'(x)^2.$$
1991 Bulgaria National Olympiad
On a unit circle with center $O$, $AB$ is an arc with the central angle $\alpha<90^\circ$. Point $H$ is the foot of the perpendicular from $A$ to $OB$, $T$ is a point on arc $AB$, and $l$ is the tangent to the circle at $T$. The line $l$ and the angle $AHB$ form a triangle $\Delta$.



(a) Prove that the area of $\Delta$ is minimal when $T$ is the midpoint of arc $AB$.

(b) Prove that if $S_\alpha$ is the minimal area of $\Delta$ then the function $\frac{S_\alpha}\alpha$ has a limit when $\alpha\to0$ and find this limit.
1991 Bulgaria National Olympiad
White and black checkers are put on the squares of an $n\times n$ chessboard $(n\ge2)$ according to the following rule. Initially, a black checker is put on an arbitrary square. In every consequent step, a white checker is put on a free square, whereby all checkers on the squares neighboring by side are replaced by checkers of the opposite colors. This process is continued until there is a checker on every square. Prove that in the final configuration there is at least one black checker.
1992 Bulgaria National Olympiad
Through a random point $C_1$ from the edge $DC$ of the regular tetrahedron $ABCD$ is drawn a plane, parallel to the plane $ABC$. The plane constructed intersects the edges $DA$ and $DB$ at the points $A_1,B_1$ respectively. Let the point $H$ is the midpoint of the altitude through the vertex $D$ of the tetrahedron $DA_1B_1C_1$ and $M$ is the center of gravity (barycenter) of the triangle $ABC_1$. Prove that the measure of the angle $HMC$ doesn’t depend on the position of the point $C_1$. (Ivan Tonov)
1992 Bulgaria National Olympiad
Prove that there exists $1904$-element subset of the set $\{1,2,\ldots,1992\}$, which doesn’t contain an arithmetic progression consisting of $41$ terms. (Ivan Tonov)
1992 Bulgaria National Olympiad
Let $m$ and $n$ are fixed natural numbers and $Oxy$ is a coordinate system in the plane. Find the total count of all possible situations of $n+m-1$ points $P_1(x_1,y_1),P_2(x_2,y_2),\ldots,P_{n+m-1}(x_{n+m-1},y_{n+m-1})$ in the plane for which the following conditions are satisfied:



(i) The numbers $x_i$ and $y_i~(i=1,2,\ldots,n+m-1)$ are integers and $1\le x_i\le n,1\le y_i\le m$.

(ii) Every one of the numbers $1,2,\ldots,n$ can be found in the sequence $x_1,x_2,\ldots,x_{n+m-1}$ and every one of the numbers $1,2,\ldots,m$ can be found in the sequence $y_1,y_2,\ldots,y_{n+m-1}$.

(iii) For every $i=1,2,\ldots,n+m-2$ the line $P_iP_{i+1}$ is parallel to one of the coordinate axes. (Ivan Gochev, Hristo Minchev)
1992 Bulgaria National Olympiad
Let $p$ be a prime number in the form $p=4k+3$. Prove that if the numbers $x_0,y_0,z_0,t_0$ are solutions of the equation $x^{2p}+y^{2p}+z^{2p}=t^{2p}$, then at least one of them is divisible by $p$. (Plamen Koshlukov)
1992 Bulgaria National Olympiad
Points $D,E,F$ are midpoints of the sides $AB,BC,CA$ of triangle $ABC$. Angle bisectors of the angles $BDC$ and $ADC$ intersect the lines $BC$ and $AC$ respectively at the points $M$ and $N$, and the line $MN$ intersects the line $CD$ at the point $O$. Let the lines $EO$ and $FO$ intersect respectively the lines $AC$ and $BC$ at the points $P$ and $Q$. Prove that $CD=PQ$. (Plamen Koshlukov)
1992 Bulgaria National Olympiad
There are given one black box and $n$ white boxes ($n$ is a random natural number). White boxes are numbered with the numbers $1,2,\ldots,n$. In them are put $n$ balls. It is allowed the following rearrangement of the balls: if in the box with number $k$ there are exactly $k$ balls, that box is made empty - one of the balls is put in the black box and the other $k-1$ balls are put in the boxes with numbers: $1,2,\ldots,k-1$. (Ivan Tonov)
1996 Bulgaria National Olympiad
Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.
1996 Bulgaria National Olympiad
Find the side length of the smallest equilateral triangle in which three discs of radii $2,3,4$ can be placed without overlap.
1996 Bulgaria National Olympiad
The quadratic polynomials $f$ and $g$ with real coefficients are such that if $g(x)$ is an integer for some $x>0$, then so is $f(x)$. Prove that there exist integers $m,n$ such that $f(x)=mg(x)+n$ for all $x$.
1996 Bulgaria National Olympiad
Sequence $\{a_n\}$ it define $a_1=1$ and

$$a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}$$for all $n\ge 1$

Prove that $\lfloor a_n^2\rfloor=n$ for all $n\ge 4.$
1996 Bulgaria National Olympiad
The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ meet each other in the point $E$, while the diagonals $AC$ and $BD$ in the point $F$. The circumcircles of the triangles $AFD$ and $BFC$ have a second common point, which is denoted by $H$. Prove that $\angle EHF=90^\circ$.
1996 Bulgaria National Olympiad
A square table of size $7\times 7$ with the four corner squares deleted is given.



What is the smallest number of squares which need to be colored black so that a $5-$square entirely uncolored Greek cross (Figure 1) cannot be found on the table?

Prove that it is possible to write integers in each square in a way that the sum of the integers in each Greek cross is negative while the sum of all integers in the square table is positive.



[asy]

size(3.5cm); usepackage("amsmath");

MP("\text{Figure }1.", (1.5, 3.5), N);

DPA(box((0,1),(3,2))^^box((1,0),(2,3)), black);

[/asy]
1997 Bulgaria National Olympiad
Consider the polynomial

$P_n(x) = \binom {n}{2}+\binom {n}{5}x+\binom {n}{8}x^2 + \cdots + \binom {n}{3k+2}x^{3k}$

where $n \ge 2$ is a natural number and $k = \left\lfloor \frac{n-2}{3} \right \rfloor$





(a) Prove that $P_{n+3}(x)=3P_{n+2}(x)-3P_{n+1}(x)+(x+1)P_n(x)$

(b) Find all integer numbers $a$ such that $P_n(a^3)$ is divisible by $3^{ \lfloor \frac{n-1}{2} \rfloor}$ for all $n \ge 2$
1997 Bulgaria National Olympiad
Let $M$ be the centroid of $\Delta ABC$

Prove the inequality

$\sin \angle CAM + \sin\angle CBM \le \frac{2}{\sqrt 3}$



(a) if the circumscribed circle of $\Delta AMC$ is tangent to the line $AB$

(b) for any $\Delta ABC$
1997 Bulgaria National Olympiad
Let $n$ and $m$ be natural numbers such that $m+ i=a_ib_i^2$ for $i=1,2, \cdots n$ where $a_i$ and $b_i$ are natural numbers and $a_i$ is not divisible by a square of a prime number.

Find all $n$ for which there exists an $m$ such that

$\sum_{i=1}^{n}a_i=12$
1997 Bulgaria National Olympiad
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$.

Prove that

$ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$.
1997 Bulgaria National Olympiad
Given a triangle $ABC$.

Let $M$ and $N$ be the points where the angle bisectors of the angles $ABC$ and $BCA$ intersect the sides $CA$ and $AB$, respectively.

Let $D$ be the point where the ray $MN$ intersects the circumcircle of triangle $ABC$.



Prove that $\frac{1}{BD}=\frac{1}{AD}+\frac{1}{CD}$.
1997 Bulgaria National Olympiad
Let $X$ be a set of $n + 1$ elements, $n\geq 2$. Ordered $n$-tuples $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ formed from distinct elements of $X$ are called disjoint if there exist distinct indices $1\leq i \neq j\leq n$ such that $a_i = b_j$. Find the maximal number of pairwise disjoint $n$-tuples.
1998 Bulgaria National Olympiad
Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of $0$ and $1$ of length $2n+2$ with the following property: any sequence of $0$ and $1$ of length $2n+2$ coincides with some of these $k$ sequences in at least $n+2$ positions.
1998 Bulgaria National Olympiad
The polynomials $P_n(x,y), n=1,2,... $ are defined by $$P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)$$ Prove that $P_{n}(x,y)=P_{n}(y,x)$ for all $x,y \in \mathbb{R}$ and $n $.
1998 Bulgaria National Olympiad
On the sides of a non-obtuse triangle $ABC$ a square, a regular $n$-gon and a regular $m$-gon ($m$,$n > 5$) are constructed externally, so that their centers are vertices of a regular triangle. Prove that $m = n = 6$ and find the angles of $\triangle ABC$.
1998 Bulgaria National Olympiad
Let $a_1,a_2,\cdots ,a_n$ be real numbers, not all zero. Prove that the equation:

$$\sqrt{1+a_1x}+\sqrt{1+a_2x}+\cdots +\sqrt{1+a_nx}=n$$

has at most one real nonzero root.
1998 Bulgaria National Olympiad
let m and n be natural numbers such that:  $3m|(m+3)^n+1$

Prove that $\frac{(m+3)^n+1}{3m}$ is odd
1998 Bulgaria National Olympiad
The sides and diagonals of a regular $n$-gon $R$ are colored in $k$ colors so that:

(i) For each color $a$ and any two vertices $A$,$B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$.

(ii) The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors.

Prove that $k\leq  2$.
1999 Bulgaria National Olympiad
The faces of a box with integer edge lengths are painted green. The box is partitioned into unit cubes. Find the dimensions of the box if the number of unit cubes with no green face is one third of the total number of cubes.
1999 Bulgaria National Olympiad
Let $\{a_n\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$. If $2000|a_{1999}$, find the smallest $n\ge 2$ such that $2000|a_n$.
1999 Bulgaria National Olympiad
The vertices of a triangle have integer coordinates and one of its sides is of length $\sqrt{n}$, where $n$ is a square-free natural number. Prove that the ratio of the circumradius and the inradius is an irrational number.
1999 Bulgaria National Olympiad
Find the number of all integers $n$ with $4\le n\le 1023$ which contain no three consecutive equal digits in their binary representations.
1999 Bulgaria National Olympiad
The vertices A,B,C of an acute-angled triangle ABC lie on the sides B1C1, C1A1, A1B1 respectively of a triangle A1B1C1 similar to the triangle ABC (∠A = ∠A1, etc.). Prove that the orthocenters of triangles ABC and A1B1C1 are equidistant from the circumcenter of △ABC.
1999 Bulgaria National Olympiad
Prove that $x^3+y^3+z^3+t^3=1999$ has infinitely many soln. over $\mathbb{Z}$.
2000 Bulgaria National Olympiad
In the coordinate plane, a set of $2000$ points $\{(x_1, y_1), (x_2, y_2), . . . , (x_{2000}, y_{2000})\}$ is called good if $0\leq x_i \leq 83$, $0\leq y_i \leq 83$ for $i = 1, 2, \dots, 2000$ and $x_i \not= x_j$ when $i\not=j$. Find the largest positive integer $n$ such that, for any good set, the interior and boundary of some unit square contains exactly $n$ of the points in the set on its interior or its boundary.
2000 Bulgaria National Olympiad
Let be given an acute triangle $ABC$. Show that there exist unique points $A_1 \in BC$, $B_1 \in CA$, $C_1 \in AB$ such that each of these three points is the midpoint of the segment whose endpoints are the orthogonal projections of the other two points on the corresponding side. Prove that the triangle $A_1B_1C_1$ is similar to the triangle whose side lengths are the medians of $\triangle ABC$.
2000 Bulgaria National Olympiad
Let $ p$ be a prime number and let $ a_1,a_2,\ldots,a_{p - 2}$ be positive integers such that $ p$ doesn't $ a_k$ or $ {a_k}^k - 1$ for any $ k$. Prove that the product of some of the $ a_i$'s is congruent to $ 2$ modulo $ p$.
2000 Bulgaria National Olympiad
Find all polynomials $P(x)$ with real coefficients such that

$$P(x)P(x + 1) = P(x^2), \quad \forall x \in \mathbb R.$$
2000 Bulgaria National Olympiad
Let $D$ be the midpoint of the base $AB$ of the isosceles acute triangle $ABC$. Choose point $E$ on segment $AB$, and let $O$ be the circumcenter of triangle $ACE$. Prove that the line through $D$ perpendicular to $DO$, the line through $E$ perpendicular to $BC$, and the line through $B$ parallel to $AC$ are concurrent.
2000 Bulgaria National Olympiad
Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.
2001 Bulgaria National Olympiad
Consider the sequence $\{a_n\}$ such that $a_0=4$, $a_1=22$, and $a_n-6a_{n-1}+a_{n-2}=0$ for $n\ge2$.  Prove that there exist sequences $\{x_n\}$ and $\{y_n\}$ of positive integers such that

$$

a_n=\frac{y_n^2+7}{x_n-y_n}

$$

for any $n\ge0$.
2001 Bulgaria National Olympiad
Suppose that $ABCD$ is a parallelogram such that $DAB>90$. Let the point $H$ to be on $AD$ such that $BH$ is perpendicular to $AD$. Let the point $M$ to be the midpoint of $AB$. Let the point $K$ to be the intersecting point of the line $DM$ with the circumcircle of $ADB$. Prove that $HKCD$ is concyclic.
2001 Bulgaria National Olympiad
Given a permutation $(a_{1}, a_{1},...,a_{n})$ of the numbers $1, 2,...,n$ one may interchange any two consecutive "blocks" - that is, one may transform

($a_{1}, a_{2},...,a_{i}$,$\underbrace {a_{i+1},... a_{i+p},}_{A} $ $ \underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n}) $

into

$ (a_{1}, a_{2},...,a_{i},$ $ \underbrace {a_{i+p+1},...,a_{i+q},}_{B} $ $ \underbrace {a_{i+1},... a_{i+p}}_{A}$$,...,a_{n}) $

by interchanging the "blocks" $A$ and $B$. Find the least number of such changes which are needed to transform $(n, n-1,...,1)$ into $(1,2,...,n)$
2001 Bulgaria National Olympiad
Let $n \geq 2$ be a given integer. At any point $(i, j)$ with $i, j \in\mathbb{ Z}$ we write the remainder of $i+j$ modulo $n$. Find all pairs $(a, b)$ of positive integers such that the rectangle with vertices $(0, 0)$, $(a, 0)$, $(a, b)$, $(0, b)$ has the following properties:

(i) the remainders $0, 1, \ldots , n-1$ written at its interior points appear the same number of times;

(ii) the remainders $0, 1, \ldots , n -1$ written at its boundary points appear the same number of times.
2001 Bulgaria National Olympiad
Find all real values $t$ for which there exist real numbers $x$, $y$, $z$ satisfying :

$3x^2 + 3xz + z^2 = 1$ ,

$3y^2 + 3yz + z^2 = 4$,

$x^2 - xy + y^2 = t$.
2001 Bulgaria National Olympiad
Let $p$ be a prime number congruent to $3$ modulo $4$, and consider the equation $(p+2)x^{2}-(p+1)y^{2}+px+(p+2)y=1$.

Prove that this equation has infinitely many solutions in positive integers, and show that if $(x,y) = (x_{0}, y_{0})$ is a solution of the equation in positive integers, then $p | x_{0}$.
2002 Bulgaria National Olympiad
Let $a_1, a_2... $ be an infinite sequence of real numbers such that $a_{n+1}=\sqrt{{a_n}^2+a_n-1}$. Prove that $a_1  \notin (-2,1)$



Proposed by Oleg Mushkarov and Nikolai Nikolov
2002 Bulgaria National Olympiad
Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$



Proposed by Alexander Ivanov
2002 Bulgaria National Olympiad
Given are $n^2$ points in the plane, such that no three of them are collinear, where $n \geq 4$ is the positive integer of the form $3k+1$. What is the minimal number of connecting segments among the points, such that for each $n$-plet of points we can find four points, which are all connected to each other?



Proposed by Alexander Ivanov and Emil Kolev
2002 Bulgaria National Olympiad
Let $I$ be the incenter of a non-equilateral triangle $ABC$ and $T_1$, $T_2$, and $T_3$ be the tangency points of the incircle with the sides $BC$, $CA$ and $AB$, respectively. Prove that the orthocenter of triangle $T_1T_2T_3$ lies on the line $OI$, where $O$ is the circumcenter of triangle $ABC$.



Proposed by Georgi Ganchev
2002 Bulgaria National Olympiad
Find all pairs $(b,c)$ of positive integers, such that the sequence defined by $a_1=b$, $a_2=c$ and $a_{n+2}= \left| 3a_{n+1}-2a_n \right|$ for $n \geq 1$ has only finite number of composite terms.



Proposed by Oleg Mushkarov and Nikolai Nikolov
2002 Bulgaria National Olympiad
Find the smallest number $k$, such that  $ \frac{l_a+l_b}{a+b}<k$ for all triangles with sides $a$ and $b$ and bisectors $l_a$ and $l_b$ to them, respectively.



Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov
2003 Bulgaria National Olympiad
Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$.