Bulgaria 1962 - 2021 120p

geometry problems from Bulgarian Mathematical Olympiads (4th/ final round)

with aops links

(in Bulgarian: Национална олимпиада по математика)

collected inside aops: here

1962 - 2021

1962 Bulgaria P2

It is given a circle with center $O$ and radius $r$. $AB$ and $MN$ are two diameters. The lines $MB$ and $NB$ are tangent to the circle at the points $M'$ and $N'$ and intersect at point $A$. $M''$ and $N''$ are the midpoints of the segments $AM'$ and $AN'$. Prove that:

(a) the points $M,N,N',M'$ are concyclic.

(b) the heights of the triangle $M''N''B$ intersect in the midpoint of the radius $OA$.

1962 Bulgaria P3

It is given a cube with sidelength $a$. Find the surface of the intersection of the cube with a plane, perpendicular to one of its diagonals and whose distance from the centre of the cube is equal to $h$.

1962 Bulgaria P4

There are given a triangle and some internal point $P$. $x,y,z$ are distances from $P$ to the vertices $A,B$ and $C$. $p,q,r$ are distances from $P$ to the sides $BC,CA,AB$ respectively. Prove that:

$$xyz\ge(q+r)(r+p)(p+q).$$

1963 Bulgaria P3

In the trapezium $ABCD$, a point $M$ is chosen on the non-base segment $AB$. Through the points $M,A,D$ and $M,B,C$ are drawn circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$. Prove that:

(a) the second intersection point $N$ of $k_1$ and $k_2$ lies on the other non-base segment $CD$ or on its continuation;

(b) the length of the line $O_1O_2$ doesn’t depend on the location of $M$ on $AB$;

(c) the triangles $O_1MO_2$ and $DMC$ are similar. Find such a position of $M$ on $AB$ that makes $k_1$ and $k_2$ have the same radius.

1963 Bulgaria P4

In the tetrahedron $ABCD$ three of the faces are right-angled triangles and the other is not an obtuse triangle. Prove that:

(a) the fourth wall of the tetrahedron is a right-angled triangle if and only if exactly two of the plane angles having common vertex with the some of vertices of the tetrahedron are equal.

(b) its volume is equal to $\frac16$ multiplied by the multiple of two shortest edges and an edge not lying on the same wall.

1964 Bulgaria P3

There are given two intersecting lines $g_1,g_2$ and a point $P$ in their plane such that $\angle(g1,g2)\ne90^\circ$. Its symmetrical points on any point $M$ in the same plane with respect to the given lines are $M_1$ and $M_2$. Prove that:

(a) the locus of the point $M$ for which the points $M_1,M_2$ and $P$ lie on a common line is a circle $k$ passing through the intersection point of $g_1$ and $g_2$.

(b) the point $P$ is an orthocenter of a triangle, inscribed in the circle $k$ whose sides lie at the lines $g_1$ and $g_2$.

1964 Bulgaria P4

Let $a_1,b_1,c_1$ are three lines each two of them are mutually crossed and aren't parallel to some plane. The lines $a_2,b_2,c_2$ intersect the lines $a_1,b_1,c_1$ at the points $a_2$ in $A$, $C_2$, $B_1$; $b_2$ in $C_1$, $B$, $A_2$; $c_2$ in $B_2$, $A_1$, $C$ respectively in such a way that $A$ is the perpendicular bisector of $B_1C_2$, $B$ is the perpendicular bisector of $C_1A_2$ and $C$ is the perpendicular bisector of $A_1B_2$. Prove that:

(a) $A$ is the perpendicular bisector of $B_2C_1$, $B$ is the perpendicular bisector of $C_2A_1$ and $C$ is the perpendicular bisector of $A_2B_1$;

(b) triangles $A_1B_1C_1$ and $A_2B_2C_2$ are the same.

1965 Bulgaria P3

In the triangle $ABC$, angle bisector $CD$ intersects the circumcircle of $ABC$ at the point $K$.

(a) Prove the equalities:

$$\frac1{ID}-\frac1{IK}=\frac1{CI},\enspace\frac{CI}{ID}-\frac{ID}{DK}=1$$where $I$ is the center of the inscribed circle of triangle $ABC$.

(b) On the segment $CK$ some point $P$ is chosen whose projections on $AC,BC,AB$ respectively are $P_1,P_2,P_3$. The lines $PP_3$ and $P_1P_2$ intersect at a point $M$. Find the locus of $M$ when $P$ moves around segment $CK$.

1965 Bulgaria P4

In the space there are given crossed lines $s$ and $t$ such that $\angle(s,t)=60^\circ$ and a segment $AB$ perpendicular to them. On $AB$ it is chosen a point $C$ for which $AC:CB=2:1$ and the points $M$ and $N$ are moving on the lines $s$ and $t$ in such a way that $AM=2BN$. The angle between vectors $\overrightarrow{AM}$ and $\overrightarrow{BM}$ is $60^\circ$. Prove that:

(a) the segment $MN$ is perpendicular to $t$;

(b) the plane $\alpha$, perpendicular to $AB$ in point $C$, intersects the plane $CMN$ on fixed line $\ell$ with given direction in respect to $s$;

(c) all planes passing by $ell$ and perpendicular to $AB$ intersect the lines $s$ and $t$ respectively at points $M$ and $N$ for which $AM=2BN$ and $MN\perp t$.

1966 Bulgaria P3

(a) In the plane of the triangle $ABC$, find a point with the following property: its symmetrical points with respect to the midpoints of the sides of the triangle lie on the circumscribed circle.

(b) Construct the triangle $ABC$ if it is known the positions of the orthocenter $H$, midpoint of the side $AB$ and the midpoint of the segment joining the feet of the heights through vertices $A$ and $B$.

1966 Bulgaria P4

It is given a tetrahedron with vertices $A,B,C,D$.

(a) Prove that there exists a vertex of the tetrahedron with the following property: the three edges of that tetrahedron through that vertex can form a triangle.

(b) On the edges $DA,DB$ and $DC$ there are given the points $M,N$ and $P$ for which:

$$DM=\frac{DA}n,\enspace DN=\frac{DB}{n+1}\enspace DP=\frac{DC}{n+2}$$where $n$ is a natural number. The plane defined by the points $M,N$ and $P$ is $\alpha_n$. Prove that all planes $\alpha_n$, $(n=1,2,3,\ldots)$ pass through a single straight line.

1967 Bulgaria P3

It is given a right-angled triangle $ABC$ and its circumcircle $k$.

(a) prove that the radii of the circle $k_1$ tangent to the cathets of the triangle and to the circle $k$ is equal to the diameter of the incircle of the triangle ABC.

(b) on the circle $k$ there may be found a point $M$ for which the sum $MA+MB+MC$ is as large as possible.

1967 Bulgaria P4

Outside of the plane of the triangle $ABC$ is given point $D$.

(a) prove that if the segment $DA$ is perpendicular to the plane $ABC$ then orthogonal projection of the orthocenter of the triangle $ABC$ on the plane $BCD$ coincides with the orthocenter of the triangle $BCD$.

(b) for all tetrahedrons $ABCD$ with base, the triangle $ABC$ with smallest of the four heights that from the vertex $D$, find the locus of the foot of that height.

1968 Bulgaria P4

On the line $g$ we are given the segment $AB$ and a point $C$ not on $AB$. Prove that on $g$, there exists at least one pair of points $P,Q$ symmetrical with respect to $C$, which divide the segment $AB$ internally and externally in the same ratios, i.e

$$\frac{PA}{PB}=\frac{QA}{QB}\qquad(1)$$If $A,B,P,Q$ are such points from the line $g$ satisfying $(1)$, prove that the midpoint $C$ of the segment $PQ$ is the external point for the segment $AB$.

1968 Bulgaria P5

The point $M$ is inside the tetrahedron $ABCD$ and the intersection points of the lines $AM,BM,CM$ and $DM$ with the opposite walls are denoted with $A_1,B_1,C_1,D_1$ respectively. It is given also that the ratios $\frac{MA}{MA_1}$, $\frac{MB}{MB_1}$, $\frac{MC}{MC_1}$, and $\frac{MD}{MD_1}$ are equal to the same number $k$. Find all possible values of $k$

1968 Bulgaria P6

Find the kind of a triangle if

$$\frac{a\cos\alpha+b\cos\beta+c\cos\gamma}{a\sin\alpha+b\sin\beta+c\sin\gamma}=\frac{2p}{9R}.$$($\alpha,\beta,\gamma$ are the measures of the angles, $a,b,c$ are the respective lengths of the sides, $p$ the semiperimeter, $R$ is the circumradius)

1969 Bulgaria P4

Find the sides of a triangle if it is known that the inscribed circle meets one of its medians in two points and these points divide the median into three equal segments and the area of the triangle is equal to $6\sqrt{14}\text{ cm}^2$.

1969 Bulgaria P6

It is given that $r=\left(3\left(\sqrt6-1\right)-4\left(\sqrt3+1\right)+5\sqrt2\right)R$ where $r$ and $R$ are the radii of the inscribed and circumscribed spheres in a regular $n$-angled pyramid. If it is known that the centers of the spheres given coincide,

(a) find $n$;

(b) if $n=3$ and the lengths of all edges are equal to a find the volumes of the parts from the pyramid after drawing a plane $\mu$, which intersects two of the edges passing through point $A$ respectively in the points $E$ and $F$ in such a way that $|AE|=p$ and $|AF|=q$ $(p<a,q<a)$, intersects the extension of the third edge behind opposite of the vertex $A$ wall in the point $G$ in such a way that $|AG|=t$ $(t>a)$.

1970 Bulgaria P4

Let $\delta_0=\triangle A_0B_0C_0$ be a triangle. On each of the sides $B_0C_0$, $C_0A_0$, $A_0B_0$, there are constructed squares in the halfplane, not containing the respective vertex $A_0,B_0,C_0$ and $A_1,B_1,C_1$ are the centers of the constructed squares. If we use the triangle $\delta_1=\triangle A_1B_1C_1$ in the same way we may construct the triangle $\delta_2=\triangle A_2B_2C_2$; from $\delta_2=\triangle A_2B_2C_2$ we may construct $\delta_3=\triangle A_3B_3C_3$ and etc. Prove that:

(a) segments $A_0A_1,B_0B_1,C_0C_1$ are respectively equal and perpendicular to $B_1C_1,C_1A_1,A_1B_1$;

(b) vertices $A_1,B_1,C_1$ of the triangle $\delta_1$ lies respectively over the segments $A_0A_3,B_0B_3,C_0C_3$ (defined by the vertices of $\delta_0$ and $\delta_1$) and divide them in ratio $2:1$.

1970 Bulgaria P5

Prove that for $n\ge5$ the side of regular inscribable $n$-gon is bigger than the side of regular $n+1$-gon circumscribed around the same circle and if $n\le4$ the opposite statement is true.

1970 Bulgaria P6

In space, we are given the points $A,B,C$ and a sphere with center $O$ and radius $1$. Find the point $X$ from the sphere for which the sum $f(X)=|XA|^2+|XB|^2+|XC|^2$ attains its maximal and minimal value. Prove that if the segments $OA,OB,OC$ are pairwise perpendicular and $d$ is the distance from the center $O$ to the centroid of the triangle $ABC$ then:

(a) the maximum of $f(X)$ is equal to $9d^2+3+6d$;

(b) the minimum of $f(X)$ is equal to $9d^2+3-6d$.

1971 Bulgaria P4

It is given a triangle $ABC$. Let $R$ be the radius of the circumcircle of the triangle and $O_1,O_2,O_3$ be the centers of excircles of the triangle $ABC$ and $q$ is the perimeter of the triangle $O_1O_2O_3$. Prove that $q\le6R\sqrt3$. When does equality hold?

1971 Bulgaria P5

Let $A_1,A_2,\ldots,A_{2n}$ are the vertices of a regular $2n$-gon and $P$ is a point from the incircle of the polygon. If $\alpha_i=\angle A_iPA_{i+n}$, $i=1,2,\ldots,n$. Prove the equality

$$\sum_{i=1}^n\tan^2\alpha_i=2n\frac{\cos^2\frac\pi{2n}}{\sin^4\frac\pi{2n}}.$$

1971 Bulgaria P6

In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$. Let $SA=m$, $SB=n$, $SC=p$, $r$ is the inradius of $ABC$. $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that

(a) $m^2+n^2+p^2\ge18r^2$;

(b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$.

1972 Bulgaria P4

Find maximal possible number of points lying on or inside a circle with radius $R$ in such a way that the distance between every two points is greater than $R\sqrt2$.

1972 Bulgaria P5

In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral.

(a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed.

(b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold?

1972 Bulgaria P6

It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that:

(a) the four altitudes of $ABCD$ intersects at a common point $H$;

(b) $AH+BH+CH+DH<p+2R$, where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$.

1973 Bulgaria P5

Let the line $\ell$ intersects the sides $AC,BC$ of the triangle $ABC$ respectively at the points $E$ and $F$. Prove that the line $\ell$ is passing through the incenter of the triangle $ABC$ if and only if the following equality is true:

$$BC\cdot\frac{AE}{CE}+AC\cdot\frac{BF}{CF}=AB.$$

1973 Bulgaria P6

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.

1974 Bulgaria P5

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.

1974 Bulgaria P6

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 $a,b,c$.

1975 Bulgaria P2

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:

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

1975 Bulgaria P4

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

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

1976 Bulgaria P1

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.

1976 Bulgaria P3

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.

1976 Bulgaria P5

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.

1977 Bulgaria P3

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 P4

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.

1977 Bulgaria P6

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.

1978 Bulgaria P4

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 P2

$k_1$ 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 $k_1$.

1978 Bulgaria P5

Prove that for every convex polygon can be found such three sequential vertices for which a circle that they lie on covers the polygon.

1978 Bulgaria P6

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 $BS>CS$.

1979 Bulgaria P2

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 P5

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.

1980 Bulgaria P2

(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 P6

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 P2

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 P3

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 P6

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 P3

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 P6

Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.

1983 Bulgaria P3

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 P4

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.

1984 Bulgaria P2

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 P6

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 P3

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 P5

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 $\angle AQB$.

1986 Bulgaria P3

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 P5

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 $k$.

1987 Bulgaria P2

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 P3

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 P5

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 $\angle BAC$.

1988 Bulgaria P3

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 P4

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.

1989 Bulgaria P1

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 P5

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.

1990 Bulgaria P2

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 P5

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 P6

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 P1

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 P5

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.

1992 Bulgaria P1

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 P5

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)

1993 Bulgaria P2

Let $M$ be an interior point of the triangle $ABC$ such that $AMC = 90^\circ$, $AMB = 150^\circ$, and $BMC = 120^\circ$. The circumcenters of the triangles $AMC$, $AMB$, and $BMC$ are $P$, $Q$, and $R$ respectively. Prove that the area of $\Delta PQR$ is greater than or equal to the area of $\Delta ABC$.

1993 Bulgaria P5

Let $Oxy$ be a fixed rectangular coordinate system in the plane.

Each ordered pair of points $A_1, A_2$ from the same plane which are different from O and have coordinates $x_1, y_1$ and $x_2, y_2$ respectively is associated with real number $f(A_1,A_2)$ in such a way that the following conditions are satisfied:

(a) If $OA_1 = OB_1$, $OA_2 = OB_2$ and $A_1A_2 = B_1B_2$ then $f(A_1,A_2) = f(B_1,B_2)$.

(b) There exists a polynomial of second degree $F(u,v,w,z)$ such that $f(A_1,A_2)=F(x_1,y_1,x_2,y_2)$.

(c) There exists such a number $\phi \in  (0,\pi)$ that for every two points $A_1, A_2$ for which $\angle A_1OA_2 = \phi$ is satisfied $f(A_1,A_2) = 0$.

(d) If the points $A_1, A_2$ are such that the triangle $OA_1A_2$ is equilateral with side $1$ then$ f(A_1,A_2) = \frac12$.

Prove that $f(A_1,A_2) = \overrightarrow{OA_1} \cdot   \overrightarrow{OA_2}$ for each ordered pair of points $A_1, A_2$.

1994 Bulgaria P1

Two circles $k_1(O_1,R)$ and $k_2(O_2,r)$ are given in the plane such that $R  \ge \sqrt2 r$ and$$O_1O_2 =\sqrt{R^2 +r^2 - r\sqrt{4R^2 +r^2}}.$$Let $A$ be an arbitrary point on $k_1$. The tangents from $A$ to $k_2$ touch $k_2$ at $B$ and $C$ and intersect $k_1$ again at $D$ and $E$, respectively. Prove that $BD \cdot CE = r^2$

1994 Bulgaria P4

Let $ABC$ be a triangle with incenter $I$, and let the tangency points of its incircle with its sides $AB$, $BC$, $CA$ be $C'$, $A'$ and $B'$ respectively. Prove that the circumcenters of $AIA'$, $BIB'$, and $CIC'$ are collinear.

1995 Bulgaria P2

Let triangle ABC has semiperimeter $ p$. E,F are located on AB such that $ CE=CF=p$. Prove that the C-excircle of triangle ABC touches the circumcircle (EFC).

1995 Bulgaria P4

Points $A_1,B_1,C_1$ are selected on the sides $BC$,$CA$,$AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$, $A_1BC_1$, $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.

1996 Bulgaria P2

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 P5

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 $\angle EHF=90^\circ$.

1997 Bulgaria P2
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 P5
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 $\frac{1}{BD}=\frac{1}{AD}+\frac{1}{CD}$.

1998 Bulgaria P3
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 $\triangle ABC$.

1999 Bulgaria P3
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 P5
The vertices $A,B,C$ of an acute-angled triangle $ABC$ lie on the sides $B_1C_1, C_1A_1, A_1B_1$ respectively of a triangle $A_1B_1C_1$ similar to the triangle $ABC$ ($\angle A = \angle A_1$, etc.). Prove that the orthocenters of triangles $AB$C and $A_1B_1C_1$ are equidistant from the circumcenter of △ABC.

2000 Bulgaria P2
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 P5

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.

2001 Bulgaria P2
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.

2002 Bulgaria P2
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 $r^2+p^2=d^2$

2002 Bulgaria P4
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 $ABC$.

2003 Bulgaria P2
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 $\angle EPC =\angle DPC$.

2004 Bulgaria P1
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$.

2005 Bulgaria P2
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 P4
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 $\widehat{EMF}$.

2006 Bulgaria P5
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.

2007 Bulgaria P1
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 $AX\cdot DY=BX\cdot CY$.

2008 Bulgaria P1

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.$

2009 Bulgaria P2
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 $A_1A', B_1B'$ and $C_1C'$ intersects at a common point.

2010 Bulgaria P2

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 P6

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 P4

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.

2012 Bulgaria P6

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 P5

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 $\triangle ABC$.

2014 Bulgaria P1

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.

2014 Bulgaria P3

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 $X$.

by Aleksandar Ivanov

2014 Bulgaria P6

Let $ABCD$ be a quadrilateral inscribed in a circle $k$. $AC$ and $BD$ meet at $E$. The rays $\overrightarrow{CB}, \overrightarrow{DA}$ meet at $F$. Prove that the line through the incenters of $\triangle ABE\,,\, \triangle ABF$ and the line through the incenters of $\triangle CDE\,,\, \triangle CDF$ meet at a point lying on the circle $k$.

2015 Bulgaria P1 

The hexagon $ABLCDK$ is inscribed and the line $LK$ intersects the segments $AD, BC, AC$ and $BD$ in points $M, N, P$ and $Q$, respectively. Prove that $NL \cdot KP \cdot MQ = KM \cdot  PN \cdot  LQ$.

2015 Bulgaria P5

In a triangle $\triangle ABC$ points $L, P$ and $Q$ lie on the segments $AB, AC$ and $BC$, respectively, and are such that $PCQL$ is a parallelogram. The circle with center the midpoint $M$ of the segment $AB$ and radius $CM$ and the circle of diameter $CL$ intersect for the second time at the point $T$. Prove that the lines $AQ, BP$ and $LT$ intersect in a point.

2016 Bulgaria P5

Let $\triangle {ABC} $ be isosceles triangle with $AC=BC$ . The point $D$ lies on the extension of $AC$ beyond $C$ and is that $AC>CD$. The angular bisector of $ \angle BCD $ intersects $BD$ at point $N$ and let $M$ be the midpoint of $BD$. The tangent at $M$ to the circumcircle of triangle $AMD$ intersects the side $BC$ at point  $P$. Prove that points $A,P,M$ and $N$ lie on a circle.

2017 Bulgaria P1

An convex qudrilateral $ABCD$ is given. $O$ is the intersection point of the diagonals $AC$ and $BD$. The points $A_1,B_1,C_1, D_1$ lie respectively on $AO, BO, CO, DO$ such that $AA_1=CC_1, BB_1=DD_1$. The circumcircles of $\triangle AOB$ and $\triangle COD$ meet at second time at $M$ and the the circumcircles of $\triangle AOD$ and $\triangle BOC$ - at $N$. The circumcircles of $\triangle A_1OB_1$ and $\triangle C_1OD_1$ meet at second time at $P$ and the the circumcircles of $\triangle A_1OD_1$ and $\triangle B_1OC_1$ - at $Q$. Prove that the quadrilateral $MNPQ$ is cyclic.

2017 Bulgaria P6

An acute non-isosceles $\triangle ABC$ is given. $CD, AE, BF$ are its altitudes. The points $E', F'$ are symetrical of $E, F$ with respect accordingly to $A$ and $B$. The point $C_1$ lies on $\overrightarrow{CD}$, such that $DC_1=3CD$. Prove that $\angle E'C_1F'=\angle ACB$

2018 Bulgaria P2

Let $ABCD$ be a cyclic quadrilateral. Let $H_{1}$ be the orthocentre of triangle $ABC$. Point $A_{1}$ is the image of $A$ after reflection about $BH_{1}$. Point $B_{1}$ is the image of of $B$ after reflection about $AH_{1}$. Let $O_{1}$ be the circumcentre of $(A_{1}B_{1}H_{1})$. Let $H_{2}$ be the orthocentre of triangle $ABD$. Point $A_{2}$ is the image of $A$ after reflection about $BH_{2}$. Point $B_{2}$ is the image of of $B$ after reflection about $AH_{2}$. Let $O_{2}$ be the circumcentre of $(A_{2}B_{2}H_{2})$. Lets denote by $\ell_{AB}$ be the line through $O_{1}$ and $O_{2}$. $\ell_{AD}$ ,$\ell_{BC}$ ,$\ell_{CD}$ are defined analogously. Let $M=\ell_{AB} \cap \ell_{BC}$, $N=\ell_{BC} \cap \ell_{CD}$, $P=\ell_{CD} \cap \ell_{AD}$,$Q=\ell_{AD} \cap \ell_{AB}$. Prove that $MNPQ$ is cyclic.

2018 Bulgaria P4

Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.

2019 Bulgaria P2

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O.$ Let the intersection points of the perpendicular bisector of $CH$ with $AC$ and $BC$ be $X$ and $Y$ respectively. Lines $XO$ and $YO$ cut $AB$ at $P$ and $Q$ respectively. If $XP+YQ=AB+XY,$ determine $\measuredangle OHC.$

2019 Bulgaria P6

A hexagon $ABCDEF$ is inscribed in a circle and $AB\cdot CD\cdot EF=BC\cdot DE\cdot FA$. The points $B$ and $B_1$ are symmetrical over the line $AC$; $D$ and $D_1$ - symmetrical over $CE$, and $F, F_1$ symmetrical over $EA$.  Prove that $\triangle BDF\sim \triangle B_1D_1F_1$.

2020 Bulgaria P1

On the sides of $\triangle{ABC}$ points $P,Q \in{AB}$ ($P$ is between $A$ and $Q$) and $R\in{BC}$ are chosen. The points $M$ and $N$ are defined as the intersection point of $AR$ with the segments $CP$ and $CQ$, respectively. If $BC=BQ$, $CP=AP$, $CR=CN$ and $\angle{BPC}=\angle{CRA}$, prove that $MP+NQ=BR$.

2021 Bulgaria P2

A point $T$ is given on the altitude through point $C$ in the acute triangle $ABC$ with circumcenter $O$, such that $\measuredangle TBA=\measuredangle ACB$. If the line $CO$ intersects side $AB$ at point $K$, prove that the perpendicular bisector of $AB$, the altitude through $A$ and the segment $KT$ are concurrent.

2021 Bulgaria P6

Point $S$ is the midpoint of arc $ACB$ of the circumscribed circle $k$ around triangle $ABC$ with $AC>BC$. Let $I$ be the incenter of triangle $ABC$. Line $SI$ intersects $k$ again at point $T$. Let $D$ be the reflection of $I$ across $T$ and $M$ be the midpoint of side $AB$. Line $IM$ intersects the line through $D$, parallel to $AB$, at point $E$. Prove that $AE=BD$.

 

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