Navvye/Olympiad-Problems-CSV huggingface dataset:
1973 Bulgaria National Olympiad
In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal.



H. Lesov
1974 Bulgaria National Olympiad
Find all natural numbers n with the following property: there exists a permutation $(i_1,i_2,\ldots,i_n)$ of the numbers $1,2,\ldots,n$ such that, if on the circular table there are $n$ people seated and for all $k=1,2,\ldots,n$ the $k$-th person is moving $i_n$ places in the right, all people will sit on different places.



V. Drenski
1974 Bulgaria National Olympiad
Let $f(x)$ and $g(x)$ be non-constant polynomials with integer positive coefficients, $m$ and $n$ are given natural numbers. Prove that there exists infinitely many natural numbers $k$ for which the numbers

$$f(m^n)+g(0),f(m^n)+g(1),\ldots,f(m^n)+g(k)$$are composite.



I. Tonov
1974 Bulgaria National Olympiad
(a) Find all real numbers $p$ for which the inequality

$$x_1^2+x_2^2+x_3^2\ge p(x_1x_2+x_2x_3)$$is true for all real numbers $x_1,x_2,x_3$.

(b) Find all real numbers $q$ for which the inequality

$$x_1^2+x_2^2+x_3^2+x_4^2\ge q(x_1x_2+x_2x_3+x_3x_4)$$is true for all real numbers $x_1,x_2,x_3,x_4$.



I. Tonov
1974 Bulgaria National Olympiad
Find the maximal count of shapes that can be placed over a chessboard with size $8\times8$ in such a way that no three shapes are not on two squares, lying next to each other by diagonal parallel $A1-H8$ ($A1$ is the lowest-bottom left corner of the chessboard, $H8$ is the highest-upper right corner of the chessboard).



V. Chukanov
1974 Bulgaria National Olympiad
Find all point $M$ lying into given acute-angled triangle $ABC$ and such that the area of the triangle with vertices on the feet of the perpendiculars drawn from $M$ to the lines $BC$, $CA$, $AB$ is maximal.



H. Lesov
1974 Bulgaria National Olympiad
In triangle pyramid $MABC$ at least two of the plane angles next to the edge $M$ are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true

$$8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1$$where $a,b,c$ are sides of triangle $ABC$ and $a_1,b_1,c_1$ are edges crossed respectively with $a,b,c$.



V. Petkov
1975 Bulgaria National Olympiad
Find all pairs of natural numbers $(m,n)$ bigger than $1$ for which $2^m+3^n$ is the square of whole number.



I. Tonov
1975 Bulgaria National Olympiad
Let $F$ be a polygon the boundary of which is a broken line with vertices in the knots (units) of a given in advance regular square network. If $k$ is the count of knots of the network situated over the boundary of $F$, and $\ell$ is the count of the knots of the network lying inside $F$, prove that if the surface of every square from the network is $1$, then the surface $S$ of $F$ is calculated with the formulae:

$$S=\frac k2+\ell-1$$

V. Chukanov
1975 Bulgaria National Olympiad
Let $f(x)=a_0x^3+a_1x^2+a_2x+a_3$ be a polynomial with real coefficients ($a_0\ne0$) such that $|f(x)|\le1$ for every $x\in[-1,1]$. Prove that



(a) there exist a constant $c$ (one and the same for all polynomials with the given property), for which

(b) $|a_0|\le4$.



V. Petkov
1975 Bulgaria National Olympiad
In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$, lying on $k$ or outside $k$. Prove that there exist infinitely many points $X$ from the given circumference for which

$$\sum_{i=1}^n A_iX^2\ge2nR^2.$$Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$, such that

$$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$

H. Lesov
1975 Bulgaria National Olympiad
Let the subbishop (a bishop is the figure moving only by a diagonal) be a figure moving only by diagonal but only in the next cells (squares) of the chessboard. Find the maximal count of subbishops over a chessboard $n\times n$, no two of which are not attacking.



V. Chukanov
1975 Bulgaria National Olympiad
Some of the faces of a convex polyhedron $M$ are painted in blue, others are painted in white and there are no two walls with a common edge. Prove that if the sum of surfaces of the blue walls is bigger than half surface of $M$ then it may be inscribed a sphere in the polyhedron given $(M)$.



(H. Lesov)
1976 Bulgaria National Olympiad
In a circle with a radius of $1$ is an inscribed hexagon (convex). Prove that if the multiple of all diagonals that connects vertices of neighboring sides is equal to $27$ then all angles of hexagon are equals.



V. Petkov, I. Tonov
1976 Bulgaria National Olympiad
Find all polynomials $p(x)$ satisfying the condition:

$$p(x^2-2x)=p(x-2)^2.$$
1976 Bulgaria National Olympiad
In the space is given a tetrahedron with length of the edge $2$. Prove that distances from some point $M$ to all of the vertices of the tetrahedron are integer numbers if and only if $M$ is a vertex of tetrahedron.



J. Tabov
1976 Bulgaria National Olympiad
Let $0<x_1\le x_2\le\ldots\le x_n$. Prove that

$$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\ge\frac{x_2}{x_1}+\frac{x_3}{x_2}+\ldots+\frac{x_n}{x_{n-1}}+\frac{x_1}{x_n}$$

I. Tonov
1976 Bulgaria National Olympiad
It is given a tetrahedron $ABCD$ and a plane $\alpha$ intersecting the three edges passing through $D$. Prove that $\alpha$ divides the surface of the tetrahedron into two parts proportional to the volumes of the bodies formed if and only if $\alpha$ is passing through the center of the inscribed tetrahedron sphere.
1976 Bulgaria National Olympiad
It is given a plane with a coordinate system with a beginning at the point $O$. $A(n)$, when $n$ is a natural number is a count of the points with whole coordinates which distances to $O$ are less than or equal to $n$.



(a) Find

$$\ell=\lim_{n\to\infty}\frac{A(n)}{n^2}.$$(b) For which $\beta$ $(1<\beta<2)$ does the following limit exist?

$$\lim_{n\to\infty}\frac{A(n)-\pi n^2}{n^\beta}$$
1977 Bulgaria National Olympiad
For natural number $n$ and real numbers $\alpha$ and $x$ satisfy the inequalities $\alpha^{n+1}\le x\le1$ and $0<\alpha<1$. Prove that

$$\prod_{k=1}^n\left|\frac{x-\alpha^k}{x+\alpha^k}\right|\le\prod_{k=1}^n\left|\frac{1-\alpha^k}{1+\alpha^k}\right|.$$

Borislav Boyanov
1977 Bulgaria National Olympiad
In the space are given $n$ points and no four of them belongs to a common plane. Some of the points are connected with segments. It is known that four of the given points are vertices of tetrahedron which edges belong to the segments given. It is also known that common number of the segments, passing through vertices of tetrahedron is $2n$. Prove that there exists at least two tetrahedrons every one of which have a common face with the first (initial) tetrahedron.



N. Nenov, N. Hadzhiivanov
1977 Bulgaria National Olympiad
A given truncated pyramid has triangular bases. The areas of the bases are $B_1$ and $B_2$ and the area of the surface is $S$. Prove that if there exists a plane parallel to the bases whose intersection divides the pyramid to two truncated pyramids in which may be inscribed by spheres then

$$S=(\sqrt{B_1}+\sqrt{B_2})(\sqrt[4]{B_1}+\sqrt[4]{B_2})^2$$

G. Gantchev
1977 Bulgaria National Olympiad
Vertices $A$ and $C$ of the quadrilateral $ABCD$ are fixed points of the circle $k$ and each of the vertices $B$ and $D$ is moving to one of the arcs of $k$ with ends $A$ and $C$ in such a way that $BC=CD$. Let $M$ be the intersection point of $AC$ and $BD$ and $F$ is the center of the circumscribed circle around $\triangle ABM$. Prove that the locus of $F$ is an arc of a circle.



J. Tabov
1977 Bulgaria National Olympiad
Let $Q(x)$ be a non-zero polynomial and $k$ be a natural number. Prove that the polynomial $P(x) = (x-1)^kQ(x)$ has at least $k+1$ non-zero coefficients.
1977 Bulgaria National Olympiad
A Pythagorean triangle is any right-angled triangle for which the lengths of two legs and the length of the hypotenuse are integers. We are observing all Pythagorean triangles in which may be inscribed a quadrangle with sidelength integer number, two of which sides lie on the cathets and one of the vertices of which lies on the hypotenuse of the triangle given. Find the side lengths of the triangle with minimal surface from the observed triangles.



St. Doduneko
1978 Bulgaria National Olympiad
We are given the sequence $a_1,a_2,a_3,\ldots$, for which:

$$a_n=\frac{a^2_{n-1}+c}{a_{n-2}}\enspace\text{for all }n>2.$$Prove that the numbers $a_1$, $a_2$ and $\frac{a_1^2+a_2^2+c}{a_1a_2}$ are whole numbers.
1978 Bulgaria National Olympiad
$k_1$ denotes one of the arcs formed by intersection of the circumference $k$ and the chord $AB$. $C$ is the middle point of $k_1$. On the half line (ray) $PC$ is drawn the segment $PM$. Find the locus formed from the point $M$ when $P$ is moving on $k_1$.



G. Ganchev
1978 Bulgaria National Olympiad
On the name day of a man there are $5$ people. The men observed that of any $3$ people there are $2$ that knows each other. Prove that the man may order his guests around circular table in such way that every man have on its both side people that he knows.



N. Nenov, N. Hazhiivanov
1978 Bulgaria National Olympiad
Find the greatest possible real value of $S$ and smallest possible value of $T$ such that for every triangle with sides $a,b,c$ $(a\le b\le c)$ to be true the inequalities:

$$S\le\frac{(a+b+c)^2}{bc}\le T.$$
1978 Bulgaria National Olympiad
Prove that for every convex polygon can be found such three sequential vertices for which a circle that they lie on covers the polygon.



Jordan Tabov
1978 Bulgaria National Olympiad
The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$. If $AS$ is the longest edge of the pyramid prove that $BS>CS$.



Jordan Tabov
1979 Bulgaria National Olympiad
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$.



Daniel Harrer
1979 Bulgaria National Olympiad
Points $P,Q,R,S$ are taken on respective edges $AC$, $AB$, $BD$, and $CD$ of a tetrahedron $ABCD$ so that $PR$ and $QS$ intersect at point $N$ and $PS$ and $QR$ intersect at point $M$. The line $MN$ meets the plane $ABC$ at point $L$. Prove that the lines $AL$, $BP$, and $CQ$ are concurrent.
1979 Bulgaria National Olympiad
Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$.
1979 Bulgaria National Olympiad
For each real number $k$, denote by $f(k)$ the larger of the two roots of the quadratic equation

$$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values.
1979 Bulgaria National Olympiad
A convex pentagon $ABCDE$ satisfies $AB=BC=CA$ and $CD=DE=EC$. Let $S$ be the center of the equilateral triangle $ABC$ and $M$ and $N$ be the midpoints of $BD$ and $AE$, respectively. Prove that the triangles $SME$ and $SND$ are similar.
1979 Bulgaria National Olympiad
The set $M=\{1,2,\ldots,2n\}~(n\ge2)$ is partitioned into $k$ nonintersecting subsets $M_1,M_2,\ldots,M_k$, where $k^3+1\le n$. Prove that there exist $k+1$ even numbers $2j_1,2j_2,\ldots,2j_{k+1}$ in $M$ that are in one and the same subset $M_j$ $(1\le j\le k)$ such that the numbers $2j_1-1,2j_2-1,\ldots,2j_{k+1}-1$ are also in one and the same subset $M_r$ $(1\le r\le k)$.
1980 Bulgaria National Olympiad
Show that there exists a unique sequence of decimal digits $p_0=5,p_1,p_2,\ldots$ such that, for any $k$, the square of any positive integer ending with $\overline{p_kp_{k-1}\cdots p_0}$ ends with the same digits.
1980 Bulgaria National Olympiad
(a) Prove that the area of a given convex quadrilateral is at least twice the area of an arbitrary convex quadrilateral inscribed in it whose sides are parallel to the diagonals of the original one.

(b) A tetrahedron with surface area $S$ is intersected by a plane perpendicular to two opposite edges. If the area of the cross-section is $Q$, prove that $S>4Q$.
1980 Bulgaria National Olympiad
Each diagonal of the base and each lateral edge of a $9$-gonal pyramid is colored either green or red. Show that there must exist a triangle with the vertices at vertices of the pyramid having all three sides of the same color.
1980 Bulgaria National Olympiad
If $a,b,c$ are arbitrary nonnegative real numbers, prove the inequality

$$a^3+b^3+c^3+6abc\ge\frac14(a+b+c)^3$$with equality if and only if two of the numbers are equal and the third one equals zero.
1980 Bulgaria National Olympiad
Prove that the number of ways of choosing $6$ among the first $49$ positive integers, at least two of which are consecutive, is equal to $\binom{49}6-\binom{44}6$.
1980 Bulgaria National Olympiad
Show that if all lateral edges of a pentagonal pyramid are of equal length and all the angles between neighboring lateral faces are equal, then the pyramid is regular.
1981 Bulgaria National Olympiad
Five points are given in space, no four of which are coplanar. Each of the segments connecting two of them is painted in white, green or red, so that all the colors are used and no three segments of the same color form a triangle. Prove that among these five points there is one at which segments of all the three colors meet.
1981 Bulgaria National Olympiad
Let $ABC$ be a triangle such that the altitude $CH$ and the sides $CA,CB$ are respectively equal to a side and two distinct diagonals of a regular heptagon. Prove that $\angle ACB<120^\circ$.
1981 Bulgaria National Olympiad
A quadrilateral pyramid is cut by a plane parallel to the base. Suppose that a sphere $S$ is circumscribed and a sphere $\Sigma$ inscribed in the obtained solid, and moreover that the line through the centers of these two spheres is perpendicular to the base of the pyramid. Show that the pyramid is regular.
1981 Bulgaria National Olympiad
Let $n$ be an odd positive integer. Prove that if the equation $\frac1x+\frac1y=\frac4n$ has a solution in positive integers $x,y$, then $n$ has at least one divisor of the form $4k-1$, $k\in\mathbb N$.
1981 Bulgaria National Olympiad
Find all positive values of $a$, for which there is a number $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points. Also prove that for every such a there exists $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points whose $x$-coordinates form an arithmetic progression.
1981 Bulgaria National Olympiad
Planes $\alpha,\beta,\gamma,\delta$ are tangent to the circumsphere of a tetrahedron $ABCD$ at points $A,B,C,D$, respectively. Line $p$ is the intersection of $\alpha$ and $\beta$, and line $q$ is the intersection of $\gamma$ and $\delta$. Prove that if lines $p$ and $CD$ meet, then lines $q$ and $AB$ lie on a plane.
1982 Bulgaria National Olympiad
Find all pairs of natural numbers $(n,k)$ for which



$(n+1)^{k}-1 = n!$.
1982 Bulgaria National Olympiad
Let $n$ unit circles be given on a plane. Prove that on one of the circles there is an arc of length at least $\frac{2\pi}n$ not intersecting any other circle.
1982 Bulgaria National Olympiad
In a regular $2n$-gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$. If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$.
1982 Bulgaria National Olympiad
If $x_1,x_2,\ldots,x_n$ are arbitrary numbers from the interval $[0,2]$, prove that

$$\sum_{i=1}^n\sum_{j=1}^n|x_i-x_j|\le n^2$$When is the equality attained?
1982 Bulgaria National Olympiad
Find all values of parameters $a,b$ for which the polynomial

$$x^4+(2a+1)x^3+(a-1)^2x^2+bx+4$$can be written as a product of two monic quadratic polynomials $\Phi(x)$ and $\Psi(x)$, such that the equation $\Psi(x)=0$ has two distinct roots $\alpha,\beta$ which satisfy $\Phi(\alpha)=\beta$ and $\Phi(\beta)=\alpha$.
1982 Bulgaria National Olympiad
Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.
1983 Bulgaria National Olympiad
Determine all natural numbers $n$ for which there exists a permutation $(a_1,a_2,\ldots,a_n)$ of the numbers $0,1,\ldots,n-1$ such that, if $b_i$ is the remainder of $a_1a_2\cdots a_i$ upon division by $n$ for $i=1,\ldots,n$, then $(b_1,b_2,\ldots,b_n)$ is also a permutation of $0,1,\ldots,n-1$.
1983 Bulgaria National Olympiad
Let $b_1\ge b_2\ge\ldots\ge b_n$ be nonnegative numbers, and $(a_1,a_2,\ldots,a_n)$ be an arbitrary permutation of these numbers. Prove that for every $t\ge0$,

$$(a_1a_2+t)(a_3a_4+t)\cdots(a_{2n-1}a_{2n}+t)\le(b_1b_2+t)(b_3b_4+t)\cdots(b_{2n-1}b_{2n}+t).$$
1983 Bulgaria National Olympiad
A regular triangular pyramid $ABCD$ with the base side $AB=a$ and the lateral edge $AD=b$ is given. Let $M$ and $N$ be the midpoints of $AB$ and $CD$ respectively. A line $\alpha$ through $MN$ intersects the edges $AD$ and $BC$ at $P$ and $Q$, respectively.



(a) Prove that $AP/AD=BQ/BC$.

(b) Find the ratio $AP/AD$ which minimizes the area of $MQNP$.
1983 Bulgaria National Olympiad
Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point.
1983 Bulgaria National Olympiad
Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have  common complex roots?
1983 Bulgaria National Olympiad
Let $a,b,c>0$ satisfy for all integers $n$, we have $$\lfloor an\rfloor+\lfloor bn\rfloor=\lfloor cn\rfloor$$Prove that at least one of $a,b,c$ is an integer.
1984 Bulgaria National Olympiad
Solve the equation $5^x7^y+4=3^z$ in nonnegative integers.
1984 Bulgaria National Olympiad
The diagonals of a trapezoid $ABCD$ with bases $AB$ and $CD$ intersect in a point $O$, and $AB/CD=k>1$. The bisectors of the angles $AOB,BOC,COD,DOA$ intersect $AB,BC,CD,DA$ respectively at $K,L,M,N$. The lines $KL$ and $MN$ meet at $P$, and the lines $KN$ and $LM$ meet at $Q$. If the areas of $ABCD$ and $OPQ$ are equal, find the value of $k$.
1984 Bulgaria National Olympiad
Points $P_1,P_2,\ldots,P_n,Q$ are given in space $(n\ge4)$, no four of which are in a plane. Prove that if for any three distinct points $P_\alpha,P_\beta,P_\gamma$ there is a point $P_\delta$ such that the tetrahedron $P_\alpha P_\beta P_\gamma P_\delta$ contains the point $Q$, then $n$ is an even number.
1984 Bulgaria National Olympiad
Let $a,b,a_2,\ldots,a_{n-2}$ be real numbers with $ab\ne0$ such that all the roots of the equation

$$ax^n-ax^{n-1}+a_2x^{n-2}+\ldots+a_{n-2}x^2-n^2bx+b=0$$are positive and real. Prove that these roots are all equal.
1984 Bulgaria National Olympiad
Let $0<x_i<1$ and $x_i+y_i=1$ for $i=1,2,\ldots,n$. Prove that

$$(1-x_1x_2\cdots x_n)^m+(1-y_1^m)(1-y_2^m)\cdots(1-y_n^m)>1$$for any natural numbers $m$ and $n$.
1984 Bulgaria National Olympiad
Let there be given a pyramid $SABCD$ whose base $ABCD$ is a parallelogram. Let $N$ be the midpoint of $BC$. A plane $\lambda$ intersects the lines $SC,SA,AB$ at points $P,Q,R$ respectively such that $\overline{CP}/\overline{CS}=\overline{SQ}/\overline{SA}=\overline{AR}/\overline{AB}$. A point $M$ on the line $SD$ is such that the line $MN$ is parallel to $\lambda$. Show that the locus of points $M$, when $\lambda$ takes all possible positions, is a segment of the length $\frac{\sqrt5}2SD$.
1985 Bulgaria National Olympiad
Let $f(x)$ be a non-constant polynomial with integer coefficients and $n,k$ be natural numbers. Show that there exist $n$ consecutive natural numbers $a,a+1,\ldots,a+n-1$ such that the numbers $f(a),f(a+1),\ldots,f(a+n-1)$ all have at least $k$ prime factors. (We say that the number $p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ has $\alpha_1+\ldots+\alpha_s$ prime factors.)
1985 Bulgaria National Olympiad
Find all real parameters $a$ for which all the roots of the equation

$$x^6+3x^5+(6-a)x^4+(7-2a)x^3+(6-a)x^2+3x+1$$are real.
1985 Bulgaria National Olympiad
A pyramid $MABCD$ with the top-vertex $M$ is circumscribed about a sphere with center $O$ so that $O$ lies on the altitude of the pyramid. Each of the planes $ACM,BDM,ABO$ divides the lateral surface of the pyramid into two parts of equal areas. The areas of the sections of the planes $ACM$ and $ABO$ inside the pyramid are in ratio $(\sqrt2+2):4$. Determine the angle $\delta$ between the planes $ACM$ and $ABO$, and the dihedral angle of the pyramid at the edge $AB$.
1985 Bulgaria National Olympiad
Seven points are given in space, no four of which are on a plane. Each of the segments with the endpoints in these points is painted black or red. Prove that there are two monochromatic triangles (not necessarily both of the same color) with no common edge. Does the statement hold for six points?
1985 Bulgaria National Olympiad
Let $P$ be a point on the median $CM$ of a triangle $ABC$ with $AC\ne BC$ and the acute angle $\gamma$ at $C$, such that the bisectors of $\angle PAC$ and $\angle PBC$ intersect at a point $Q$ on the median $CM$. Determine $\angle APB$ and $\angle AQB$.
1985 Bulgaria National Olympiad
Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit.
1986 Bulgaria National Olympiad
Find the smallest natural number $n$ for which the number $n^2-n+11$ has exactly four prime factors (not necessarily distinct).
1986 Bulgaria National Olympiad
Let $f(x)$ be a quadratic polynomial with two real roots in the interval $[-1,1]$. Prove that if the maximum value of $|f(x)|$ in the interval $[-1,1]$ is equal to $1$, then the maximum value of $|f'(x)|$ in the interval $[-1,1]$ is not less than $1$.
1986 Bulgaria National Olympiad
A regular tetrahedron of unit edge is given. Find the volume of the maximal cube contained in the tetrahedron, whose one vertex lies in the feet of an altitude of the tetrahedron.
1986 Bulgaria National Olympiad
Find the smallest integer $n\ge3$ for which there exists an $n$-gon and a point within it such that, if a light bulb is placed at that point, on each side of the polygon there will be a point that is not lightened. Show that for this smallest value of $n$ there always exist two points within the $n$-gon such that the bulbs placed at these points will lighten up the whole perimeter of the $n$-gon.
1986 Bulgaria National Olympiad
Let $A$ be a fixed point on a circle $k$. Let $B$ be any point on $k$ and $M$ be a point such that $AM:AB=m$ and $\angle BAM=\alpha$, where $m$ and $\alpha$ are given. Find the locus of point $M$ when $B$ describes the circle $k$.
1986 Bulgaria National Olympiad
Let $0<k<1$ be a given real number and let $(a_n)_{n\ge1}$ be an infinite sequence of real numbers which satisfies $a_{n+1}\le\left(1+\frac kn\right)a_n-1$. Prove that there is an index $t$ such that $a_t<0$.
1987 Bulgaria National Olympiad
Let $f(x)=x^n+a_1x^{n-1}+\ldots+a_n~(n\ge3)$ be a polynomial with real coefficients and $n$ real roots, such that $\frac{a_{n-1}}{a_n}>n+1$. Prove that if $a_{n-2}=0$, then at least one root of $f(x)$ lies in the open interval $\left(-\frac12,\frac1{n+1}\right)$.
1987 Bulgaria National Olympiad
Let there be given a polygon $P$ which is mapped onto itself by two rotations: $\rho_1$ with center $O_1$ and angle $\omega_1$, and $\rho_2$ with center $O_2$ and angle $\omega_2~(0<\omega_i<2\pi)$. Show that the ratio $\frac{\omega_1}{\omega_2}$ is rational.
1987 Bulgaria National Olympiad
Let $MABCD$ be a pyramid with the square $ABCD$ as the base, in which $MA=MD$, $MA^2+AB^2=MB^2$ and the area of $\triangle ADM$ is equal to $1$. Determine the radius of the largest ball that is contained in the given pyramid.
1987 Bulgaria National Olympiad
The sequence $(x_n)_{n\in\mathbb N}$ is defined by $x_1=x_2=1$, $x_{n+2}=14x_{n+1}-x_n-4$ for each $n\in\mathbb N$. Prove that all terms of this sequence are perfect squares.