Navvye/Olympiad-Problems-CSV huggingface dataset:
Brazil NMO L2 (OBM) - geometry
Let $ABCD$ be a convex quadrilateral. Let $E$ be the intersection of line $AB$ with the line $CD$, and $F$ is the intersection of line $BC$ with the line $AD$.

Let $P$ and $Q$ be the foots of the perpendicular of $E$ to the lines $AD$ and $BC$ respectively, and let $R$ and $S$ be the foots of the perpendicular of $F$ to the lines $AB$ and $CD$, respectively.The point $T$ is the intersection of the line $ER$ with the line $FS$.

a) Show that, there exists a circle that passes in the points $E, F, P, Q, R$ and $S$.

b)Show that, the circumcircle of triangle $RST$ is tangent with the circumcircle of triangle $QRB$.
Brazil NMO L2 (OBM) - geometry
Let $ABC$ a scalene triangle and $AD, BE, CF$ your angle bisectors, with $D$ in the segment $BC, E$ in the segment $AC$ and $F$ in the segment $AB$. If $\angle AFE = \angle ADC$.

Determine $\angle BCA$.
Brazil NMO L2 (OBM) - geometry
The inner bisections of the $ \angle ABC $ and $ \angle ACB $ angles of the $ ABC $ triangle are at $ I $. The $ BI $ parallel line that runs through the point $ A $ finds the $ CI $ line at the point $ D $. The $ CI $ parallel line for $ A $ finds the $ BI $ line at the point $ E $. The lines $ BD $ and $ CE $ are at the point $ F $. Show that $ F, A $, and $ I $ are collinear if and only if $ AB = AC. $
Brazil NMO L2 (OBM) - geometry
Consider a scalene triangle $ ABC $ with $ AB <AC <BC. $ The $ AB $ side mediator cuts the $ B $ side at the $ K $ point and the $ AC $ prolongation at the $ U. $ point. $ AC $ side cuts $ BC $ side at $ O $ point and $ AB $ side extension at $ G$ point.  Prove that the $ GOKU $ quad is cyclic, meaning its four vertices are at same circumference
Brazil NMO L2 (OBM) - geometry
The points $X, Y,Z$ are marked on the sides $AB, BC,AC$ of the triangle $ABC$, respectively. Points $A',B', C'$ are on the $XZ, XY, YZ$ sides of the triangle $XYZ$, respectively, so that $\frac{AB}{A'B'} = \frac{AB}{A'B'} =\frac{BC}{B'C'}= 2$ and $ABB'A',BCC'B',ACC'A'$ are trapezoids in which the sides of the triangle $ABC$ are bases.

a) Determine the ratio between the area of the trapezium $ABB'A'$ and the area of the triangle $A'B'X$.

b) Determine the ratio between the area of the triangle  $XYZ$ and the area of the triangle $ABC$.
Brazil NMO L2 (OBM) - geometry
Let $ABC$ be a triangle with  $AB \ne AC$, and let $K$ be the incenter. Let $P$ and $Q$ be the other points of the intersections of the circumcircle of triangle $BCK$  with the lines $AB$ and $AC$, respectively. Let $D$ be intersection of $AK$ and $BC$.

a) Show that $P, Q, D$ are collinear,

b) Let $T$, diiferent from $P$, be the intersection of the circumcircles of triangles $PQB$ and $QDC$. Prove that $T$ lies on the circumcircle of the triangle $ABC$.



[full wording with source: 2017 Brazil MO level 2 p5 - OBM - above was only the 1st part]
Brazil NMO L2 (OBM) - geometry
a) In $XYZ$ triangle, the incircle touches $XY$ and $XZ$ in $T$ and $W$, respectively. Prove that:

$$XT=XW=\frac{XY+XZ-YZ}2$$



Let $ABC$ a triangle and $D$ the foot of the perpendicular of $A$ in $BC$. Let $I$, $J$ be the incenters of $ABD$ and $ACD$, respectively. The incircles of $ABD$ and $ACD$ touch $AD$ in $M$ and $N$, respectively. Let $P$ be where the incircle of $ABC$ touches $AB$. The circle with centre $A$ and radius $AP$ intersects $AD$ in $K$.



b) Show that $\triangle IMK \cong \triangle KNJ$.

c) Show that $IDJK$ is cyclic.
1962 Bulgaria National Olympiad
It is given the expression $y=\frac{x^2-2x+1}{x^2-2x+2}$, where $x$ is a variable. Prove that:



(a) if $x_1$ and $x_2$ are two values of $x$, the $y_1$ and $y_2$ are the respective values of $y$ only if $x_1<x_2$, $y_1<y_2$;

(b) when $x$ is varying $y$ attains all possible values for which $0\le y<1$.
1962 Bulgaria National Olympiad
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 National Olympiad
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 National Olympiad
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 National Olympiad
Find all three-digit numbers whose remainders after division by $11$ give quotient, equal to the sum of the squares of its digits.
1963 Bulgaria National Olympiad
It is given the equation $x^2+px+1=0$, with roots $x_1$ and $x_2$;



(a) find a second-degree equation with roots $y_1,y_2$ satisfying the conditions $y_1=x_1(1-x_1)$, $y_2=x_2(1-x_2)$;

(b) find all possible values of the real parameter $p$ such that the roots of the new equation lies between $-2$ and $1$.
1963 Bulgaria National Olympiad
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 National Olympiad
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 National Olympiad
A $6n$-digit number is divisible by $7$. Prove that if its last digit is moved to the beginning of the number then the new number is also divisible by $7$.
1964 Bulgaria National Olympiad
Find all $n$-tuples of reals $x_1,x_2,\ldots,x_n$ satisfying the system:

$$\begin{cases}x_1x_2\cdots x_n=1\\x_1-x_2x_3\cdots x_n=1\\x_1x_2-x_3x_4\cdots x_n=1\\\vdots\\x_1x_2\cdots x_{n-1}-x_n=1\end{cases}$$
1964 Bulgaria National Olympiad
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 National Olympiad
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 National Olympiad
The numbers $2,3,7$ have the property that the product of any two of them increased by $1$ is divisible by the third number. Prove that this triple of integer numbers greater than $1$ is the only triple with the given property.
1965 Bulgaria National Olympiad
Prove the inequality:

$$(1+\sin^2\alpha)^n+(1+\cos^2\alpha)^n\ge2\left(\frac32\right)^n$$is true for every natural number $n$. When does equality hold?
1965 Bulgaria National Olympiad
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 National Olympiad
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 National Olympiad
Prove that the equation

$$3x(x-3y)=y^2+z^2$$doesn't have any integer solutions except $x=0,y=0,z=0$.
1966 Bulgaria National Olympiad
Prove that for every four positive numbers $a,b,c,d$ the following inequality is true:

$$\sqrt{\frac{a^2+b^2+c^2+d^2}4}\ge\sqrt[3]{\frac{abc+abd+acd+bcd}4}.$$
1966 Bulgaria National Olympiad
(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 National Olympiad
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 National Olympiad
The numbers $12,14,37,65$ are one of the solutions of the equation $xy-xz+yt=182$. What number corresponds to which letter?
1967 Bulgaria National Olympiad
Prove that:

(a) if $y<\frac12$ and $n\ge3$ is a natural number then $(y+1)^n\ge y^n+(1+2y)^\frac n2$;

(b) if $x,y,z$ and $n\ge3$ are natural numbers for which $x^2-1\le2y$ then $x^n+y^n\ne z^n$.
1967 Bulgaria National Olympiad
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 National Olympiad
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 National Olympiad
Find all natural values of $k$ for which the system

$$\begin{cases}x_1+x_2+\ldots+x_k=9\\\frac1{x_1}+\frac1{x_2}+\ldots+\frac1{x_k}=1\end{cases}$$has solutions in positive numbers. Find these solutions.



I. Dimovski
1968 Bulgaria National Olympiad
Find all functions $f:\mathbb R\to\mathbb R$ satisfying the inequality

$$xf(y)+yf(x)=(x+y)f(x)f(y)$$for all reals $x,y$. Prove that exactly two of them are continuous.



I. Dimovski
1968 Bulgaria National Olympiad
Prove that a binomial coefficient $\binom nk$ is odd if and only if all digits $1$ of $k$, when $k$ is written in binary, are on the same positions when $n$ is written in binary.



I. Dimovski
1968 Bulgaria National Olympiad
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 $AB$.



K. Petrov
1968 Bulgaria National Olympiad
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 $k$.



K. Petrov
1968 Bulgaria National Olympiad
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)



K. Petrov
1969 Bulgaria National Olympiad
Prove that if the sum of $x^5,y^5$ and $z^5$, where $x,y$ and $z$ are integer numbers, is divisible by $25$ then the sum of some two of them is divisible by $25$.
1969 Bulgaria National Olympiad
Prove that

$$S_n=\frac1{1^2}+\frac1{2^2}+\ldots+\frac1{n^2}<2$$for every $n\in\mathbb N$.
1969 Bulgaria National Olympiad
Some of the points in the plane are white and some are blue (every point of the plane is either white or blue). Prove that for every positive number $r$:



(a) there are at least two points with different color such that the distance between them is equal to $r$;

(b) there are at least two points with the same color and the distance between them is equal to $r$;

(c) will the statements above be true if the plane is replaced with the real line?
1969 Bulgaria National Olympiad
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 National Olympiad
Prove the equality

$$\prod_{k=1}^{2m}\cos\frac{k\pi}{2m+1}=\frac{(-1)^m}{4m}.$$
1969 Bulgaria National Olympiad
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 National Olympiad
Find all natural numbers $a>1$, with the property that every prime divisor of $a^6-1$ divides also at least one of the numbers $a^3-1$, $a^2-1$.



K. Dochev
1970 Bulgaria National Olympiad
Two bicyclists traveled the distance from $A$ to $B$, which is $100$ km, with speed $30$ km/h and it is known that the first started $30$ minutes before the second. $20$ minutes after the start of the first bicyclist from $A$, there is a control car started whose speed is $90$ km/h and it is known that the car is reached the first bicyclist and is driving together with him for $10$ minutes, went back to the second and was driving for $10$ minutes with him and after that the car is started again to the first bicyclist with speed $90$ km/h and etc. to the end of the distance. How many times will the car drive together with the first bicyclist?



K. Dochev
1970 Bulgaria National Olympiad
On a chessboard (with $64$ squares) there are situated $32$ white and $32$ black pools. We say that two pools form a mixed pair when they are with different colors and they lie on the same row or column. Find the maximum and the minimum of the mixed pairs for all possible situations of the pools.



K. Dochev
1970 Bulgaria National Olympiad
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 $2:1$.



K. Dochev
1970 Bulgaria National Olympiad
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 National Olympiad
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 $9d^2+3-6d$.



K. Dochev and I. Dimovski
1971 Bulgaria National Olympiad
A natural number is called triangular if it may be presented in the form $\frac{n(n+1)}2$. Find all values of $a$ $(1\le a\le9)$ for which there exist a triangular number all digit of which are equal to $a$.
1971 Bulgaria National Olympiad
Prove that the equation

$$\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$$has no real solutions.
1971 Bulgaria National Olympiad
There are given $20$ points in the plane, no three of which lie on a single line. Prove that there exist at least $969$ quadrilaterals with vertices from the given points.
1971 Bulgaria National Olympiad
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 National Olympiad
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 National Olympiad
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 National Olympiad
Prove that there are don't exist integers $a,b,c$ such that for every integer $x$ the number $A=(x+a)(x+b)(x+c)-x^3-1$ is divisible by $9$.



I. Tonov
1972 Bulgaria National Olympiad
Solve the system of equations:

$$\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\\\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\\\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt z\\x+y+z=2t\end{cases}$$if the following conditions are satisfied: $0<x<t$, $0<y<t$, $0<z<t$.



H. Lesov
1972 Bulgaria National Olympiad
Prove the equality:

$$\sum_{k=1}^{n-1}\frac1{\sin^2\frac{(2k+1)\pi}{2n}}=n^2$$where $n$ is a natural number.



H. Lesov
1972 Bulgaria National Olympiad
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 $R\sqrt2$.



H. Lesov
1972 Bulgaria National Olympiad
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?



H. Lesov
1972 Bulgaria National Olympiad
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 $ABCD$.



H. Lesov
1973 Bulgaria National Olympiad
Let the sequence $a_1,a_2,\ldots,a_n,\ldots$ is defined by the conditions: $a_1=2$ and $a_{n+1}=a_n^2-a_n+1$ $(n=1,2,\ldots)$. Prove that:



(a) $a_m$ and $a_n$ are relatively prime numbers when $m\ne n$.

(b) $\lim_{n\to\infty}\sum_{k=1}^n\frac1{a_k}=1$



I. Tonov
1973 Bulgaria National Olympiad
Let the numbers $a_1,a_2,a_3,a_4$ form an arithmetic progression with difference $d\ne0$. Prove that there are no exists geometric progressions $b_1,b_2,b_3,b_4$ and $c_1,c_2,c_3,c_4$ such that:

$$a_1=b_1+c_1,a_2=b_2+c_2,a_3=b_3+c_3,a_4=b_4+c_4.$$
1973 Bulgaria National Olympiad
Let $a_1,a_2,\ldots,a_n$ are different integer numbers in the range $[100,200]$ for which: $a_1+a_2+\ldots+a_n\ge11100$. Prove that it can be found at least number from the given in the representation of decimal system on which there are at least two equal digits.



L. Davidov
1973 Bulgaria National Olympiad
Find all functions $f(x)$ defined in the range $\left(-\frac\pi2,\frac\pi2\right)$ that are differentiable at $0$ and satisfy

$$f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)$$for every $x$ in the range $\left(-\frac\pi2,\frac\pi2\right)$.



L. Davidov
1973 Bulgaria National Olympiad
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:

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

H. Lesov