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Moscow City Oral Team MO 1999 - 2021 VIII-XI 248p

geometry problems from Oral Moscow City Team Mathematical Olympiad / Moscow Tournament of Mathematical Battles with aops links in the names  

As in math battles, the team receives a list of problems and solves them within four hours. As the solution progresses, you can tell the solved problems to the jury members. Three attempts are given for each task. The problem is either accepted (if it is completely solved) or not accepted at all (partial solutions are not evaluated).

collected inside aops: here



1999-2021


$4$ straight lines are given. Construct a square with vertices on these lines.

A convex quadrilateral can be cut into two equal polygons. Does this quadrilateral necessarily have a center or axis of symmetry?

We call a diagonal of a pentagon good if it divides its area in half. What is the largest number of good diagonals a convex pentagon can have?

Point $O$ was taken in square $ABCD$ . Prove that the centroids of triangles $AOB , BOC , COD , DOA$ form a square.

In triangle $ABC, A', B', C'$ are the points of tangency of sides with the incircle, $M$ is the midpoint of $A'B'$ . Prove that the angle $AMB$ is obtuse.

Six circles have a non-empty intersection. Prove that among them there is a circle, inside which lies the center of one of the remaining circles.

Given a tetrahedron $A_1 A_2 A_3 A_4$ . $M_i$ is the center of gravity of the face opposite $A_i , H_i$ is its orthocenter. Prove that the lines passing through $H_i$ and perpendicular to the corresponding planes of the faces of the tetrahedron intersect at one point if and only if this property is possessed by the perpendiculars passing through $M_i$ .

A segment bounded by an arc and a chord $AB$ contains a circle tangent to the arc at point $C$, and the chords at point $D$. Prove that $CD$ is a bisector of angle $ACB$.

Prove that for any tetrahedron it is possible to construct a triangle whose sides are some three edges going out from its vertex.

Three equal circles touch each other. From an arbitrary point of the fourth circle tangent to them are drawn externally to all these circles. Prove that the sum of the lengths of two tangents is equal to the length of the third.

Right-angled triangle $ABC$ moves along a plane so that the vertices $A$ and $B$ of its acute corners slide along the sides of this right angle. Prove that the set of points $C$ is a line segment and find its length.

Two crossing perpendicular lines are given in space. Find the set of midpoints of all segments of a given length whose ends lie on these lines.

Three vertices of cells $A, B, C$ are selected on checkered paper, and triangle $ABC$ is acute-angled. Prove that inside it there is at least one more vertex of the cell.

Given a convex pentagon $ABCDE$. The sides opposite to the vertices $A, B, C, D, E$ are called $CD, DE, EA, AB, BC$, respectively. A straight line passing through the vertex of the pentagon is called good if it intersects the opposite side. An arbitrary point inside the pentagon is connected by straight lines to all its vertices. Prove that there are an odd number of good lines among these lines.

Prove that if the opposite sides of the hexagon are parallel, and the diagonals connecting the opposite vertices are equal, then a circle can be circumscribed around it.

A circle rolls along the side of a an equilateral triangle, the radius of which is equal to the height of this triangle. Prove that the sides of a triangle cut out all the time on the circle arc $60^o$ .

Let's call the device marking its midpoint on the segment as a "half-meter". Using a half-meter and a ruler, divide the line into $3$ equal parts.

The centers of three pairwise tangent circles form a right-angled triangle of perimeter $p$. Find the radius of the circle that touches the three given and contains it inside itself.

Points $A$ and $B$ move in the plane along two intersecting straight lines at the same speeds. Is there a fixed point in this plane that is equidistant from $A$ and $B$ at any time?

In triangle $ABC$ medians $AA_1, BB_1, CC_1$ and altitude $AA_2, BB_2, CC_2$ are drawn. Prove that the perimeter of the polyline $A_1B_2C_1A_2B_1C_2A_1$ equals the perimeter of the triangle.

Another convex polygon is inscribed in a convex polygon (the vertices of the second lie on the sides of the first, one vertex on each side). These polygons are inscribed with circles with radii $R_1$ and $R_2$, respectively ($R_1> R_2$). Each of the triangles formed at the vertices of the first polygon has an inscribed radius $r$. Prove that $R_1 = R_2 + r$.

At the base of the pyramid lies a parallelogram. Prove that you can make a tetrahedron from its lateral faces, and the volume of this tetrahedron will be half the volume of the original pyramid.

Given a triangle $ABC$. Find the locus of points $P$ such that the sum $PA^2 + PB^2 + PC^2$ is constant.

 The ant crawls on the surface of a cube with an edge of $1$ cm. It wants to visit all the edges of the cube and return to the starting point. Find the smallest possible path length.

Vasya marked a dot on the plane with sympathetic ink and drew a square with ordinary ink. Petya sees a square, but does not see a point. He can draw a straight line on the plane and ask Vasya on which side of the straight line the point is. How many questions does he need to know if a point is inside or outside the square?

Trapezoids are made from four segments of different lengths. What is the largest number of non congruent trapezoids you can get?

In triangle $ABC$, on the median $BM$, we took point $D$ and built a triangle $CDE$, in which $DE \parallel AB$, $CE \parallel BM$. Prove that $AD = BE$.

Given a parallelogram $ABCD$. Points $H$ and $K$ are chosen on lines $AB$ and $BC$, respectively, so that $KA = AB$ and $HC = CB$. Prove that triangle $KDH$ is isosceles.

Segments $AB, CD, EF$ intersect at one point. Point $E$ belongs to segment $AC$, and point $F$ belongs to segment $BD$. Prove that $EF$ is at least one of the line segments $AB$ and $CD$.

Prove the three-dimensional cosine theorem: the square of the area of the base of the tetrahedron is equal to the sum of the squares of the areas of the side faces minus the sum of the doubled pairwise products of the areas of the side faces by the cosines of the angles between them.

The houses of Winnie the Pooh and his eight friends are located at the vertices of a convex polygon. Rabbit lives farthest from the house of Winnie the Pooh - $750$ meters to his house. Can Winnie the Pooh bypass all his friends and return home, while walking less than $4$ km?

Excircles of triangle $ABC$ touch its sides at points $A ', B', C '$. Point $A$ lies on the circle circumscribed around triangle $A'B'C '$. Prove that the second point of intersection of this circle with side $BC$ is the base of the altitude dropped to this side.

Find the volume of a body consisting of all points whose distance from the surface of the unit cube does not exceed $1$.

On the side $AC$ of triangle $ABC$, two different points $K$ and $M$ are chosen so that each of the segments $BK$ and $BM$ divides triangle $ABC$ into two isosceles triangles. Find the angles of triangle $ABC$.

Four grasshoppers sit at the vertices of a square. Every minute one of them jumps to a point symmetrical to him with respect to the other grasshopper. Prove that grasshoppers cannot end up at the vertices of a larger square at some point.

In parallelogram $ABCD$, a circle is circumscribed near triangle $ABC$. The bisector of angle $D$ intersects this circle outside the parallelogram at point $K$, different from point $B$. Prove that line $BK$ cannot be parallel to line $AC$.

All faces of a convex polyhedron are parallelograms. Prove that their number is $n (n + 1)$ for some $n$.

The square $EFGH$ is inscribed in the quadrilateral $ABCD$ such that $E$ lies on $AB, F$ lies on $BC, G$ lies on $CD$, and $H$ lies on $DA$. Prove that if $BE = CF = DG = AH$ then $ABCD$ is a square.

You are given $2$ quadrangles $ABCD$ and $A_1B_1C_1D_1$. It is known that $AB = A_1B_1 = a$, $BC = B_1C_1 = b$, $CD = C_1D_1 = c$, $DA = D_1A_1 = d$, and $AC \perp BD$. Is it mandatory $A_1C_1 \perp B_1D_1$?

The medians divide triangle $ABC$ into six triangles. It turned out that four of the circles inscribed in these triangles are equal. Prove that triangle $ABC$ is regular.

Which of the polygons inscribed in a given circle has the greatest sum of squares of its sides?

We will call a Chevian of a triangle any segment connecting one of its vertices with a point on the opposite side or its extension. Prove that for any acute-angled triangle in space, there is a point from which any of its chevians is visible at right angles.

The circle, centered on the larger base of the trapezoid, touches the other three sides. Prove that this base is equal to the sum of the two legs of the trapezoid.

Two opposite sides of a convex quadrilateral lie on perpendicular lines. Prove that the distance between the midpoints of the other two sides of the quadrilateral is equal to the distance between the midpoints of its diagonals.

A non-isosceles triangle $ABC$ is given. Points $A_1, B_1, C_1$ are the midpoints of the sides $BC, AC$ and $AB$, respectively. Points $A_2, B_2, C_2$ are the points of tangency of the incircle of this triangle, respectively. Points $A_3, B_3$ and $C_3$ are points symmetric of the points $A_2, B_2$ and $C_2$ wrt the bisector of the opposite angle ($A_2$ and $A_3$ are symmetric relative to the bisector of angle $A$, etc.) Prove that straight lines $A_1A_3, B_1B_3$ and $C_1C_3$ intersect at one point.

In an isosceles right-angled triangle $ABC$ ($AB = BC$), medians $AD$ and $CE$ were drawn. Point $X$ lies on the extension of median $AD$ beyond point $D$ such that $AD=DX$. Point $Y$ lies on the extension of median $CE$ was extended beyond point $C$ such that $CE=EY$. Prove that angle $AXY$ is right.

The vertices of one parallelogram are located on the sides of the other (one on each side). Prove that the centers of the parallelograms coincide.

Given an acute angle and on its side point $A$. Where is point $M$ on this side, which is equidistant from point $A$ and from the other side of the angle?

Is it true that for any point inside a convex quadrilateral the sum of the distances from it to the vertices is less than the perimeter of the quadrilateral?

The radius $OP$ of the circle with center $O$ intersects the perpendicular bisector of the chord $AB$ at the point $Q$. Through some point of the circle $C$, there are lines $CP$, intersecting $AB$ at a single point $X$, and $CQ$, intersecting the circle at a single point $Y$ Prove that $PX> QY$.

What is the largest of the volumes of the parallelepipeds located inside the tetrahedron of volume $1$?

What is the largest value that the length of the segment cut by the sides of the triangle on the tangent to the inscribed circle parallel to the base can take if the perimeter of the triangle is $2p$?

Point $O$ is the midpoint of the altitude of the regular tetrahedron $ABCD$. All kinds of straight lines are drawn through it, the segments of which, enclosed within the tetrahedron, are divided by the point $O$ in half. What set do the ends of these segments form on the surface of the tetrahedron?

Does there exist a convex quadrilateral with equal diagonals such that the perpendicular bisector to any of its sides does not intersect the opposite side?

In a convex quadrilateral, three sides are equal, and in the middle of the fourth point $M$ is taken. It turned out that from this point the opposite side is visible at a right angle. Find the angle between the diagonals of this quadrilateral.

A rectangle with a common right angle is inscribed in a right-angled triangle. Prove that the area of the rectangle is at most half the area of the triangle.

In an equilateral triangle $ABC$ from point $O$ on the base of $BC$, perpendiculars $OK$ (on $AB$) and $OM$ (on $AC$) are drawn, $D$ is the midpoint of $BC$. Prove that the perimeter of the quadrilateral $AMOK$ is equal to the perimeter of the triangle $ACD$.

Points $K$ and $L$ are marked on the bisector of angle $A$ of triangle $ABC$ so that $\angle ABK = \angle ACL = 90^o$. Prove that the midpoint of the line segment $KL$ is equidistant from points $B$ and $C$.

Is there a non-right non-isosceles triangle, the lengths of all sides and altitudes of which are integer?
Point $P$ lies inside triangle $ABC$. Prove that at least one of the line segments $PA, PB$ and $PC$ is not longer than the the radius circumscribed circle of the triangle $ABC$.

Let $ABCD$ be an inscribed quadrilateral. Prove that the centers of the circles inscribed in triangles $ABD, ABC, BCD$ and $ACD$ are the vertices of a rectangle.

There are several straight lines on the plane, any two of which intersect. Prove that if at least three given lines pass through any intersection point, then all lines pass through one point.

In tetrahedron $ABCD$, the dihedral angle at the edge $AB$ is equal to the dihedral angle at the edge $CD$, the dihedral angle at the edge $BC$ is equal to the dihedral angle at the edge $AD$. Prove that $S_{ABC}= S_{ADC}$

Convex hexagon $ABCDEF$ is inscribed in a circle. Prove that its diagonals $AD, BE$ and $CF$ meet at one point if and only if $AB \cdot CD \cdot   EF = BC\cdot  DE \cdot FA$.

"Half-size" allows you to draw a straight line through a given point of the plane, dividing the area of a given convex figure in half. Is it possible to divide a random angle into three equal parts using a half-size, a compass and a ruler?

What can be the angle $B$ in triangle $ABC$ if the distance between the feet of the altitudes dropped from the vertices $A$ and $C$ is equal to half the radius of the circle circumscribed about triangle $ABC$?

Find the point inside the triangle for which the product of the distances from it to the lines containing the sides of the triangle is greatest.

At the base of the triangular pyramid $ABCD$ lies an equilateral triangle $ABC$ . It is known that $AD = BC$ , and all flat angles at the vertex $D$ are equal to each other. What can these angles be equal to?

Prove that if the sides of a triangle $a , b , c$ are related by $a <( b + c ) / 2$, then the opposite angles $\angle A,\angle B,\angle C$ are related by the inequality $\angle A<(\angle B + \angle C ) / 2$,

On the diagonal $AC$ of the square $ABCD$, we took point $O$, equidistant from the vertex $D$ and the midpoint of the side $BC$. In what ratio does it, divide the diagonal?

Businessman Vladimir Petrovich has a quadrangular lawn. Vladimir Petrovich amuses himself by walking on the lawn and at each point measuring the sum of the distances to the four borders of the lawn. Each time the result of his measurements is the same. Is it true that the lawn has a parallelogram shape?

Prove that any parallelogram whose altitude is equal to the base can be cut into parts from which you can add together in order to create a square of the same area.

In a convex polygon, we chose a point and dropped the perpendiculars to the lines containing the sides of the polygon. Could it be that all the bases of the perpendiculars fall on the extensions of the sides?

In triangle $ABC$, point $D$ is the midpoint of side $AC$, point $E$ lies on side $BC$, and angle $AEB$ is equal to angle $DEC$. What is the ratio of $AE$ to $ED$?

On side $AB$ of triangle $ABC$, in which $\angle BAC = \angle BCA = 80^o$, point $D$ is taken so that $BD = AC$. Find $\angle ADC$.

Given a triangle $ABC$ with $AC = BC$. Find the locus of points $M$ such that $\angle AMC =\angle BMC$.

$H$ is the orthocenter of a non-isosceles triangle $ABC$. Let $M$ be the midpoint of $BC$, $A_1$ be the intersection point of the line $AM$ with the circumscribed circle of triangle $ABC$, $A_2$ be the symmetric to point $A_1$ wrt to $M$. Prove that $A_2H$ is perpendicular to $AM$.

A circle inscribed in triangle $ABC$ touches its sides at points $A', B', C'$. Prove that the product of the lengths of the perpendiculars dropped from any point of the circle to the sides of the triangle $ABC$ is equal to the product of the lengths of the perpendiculars dropped from the same point to the sides of the triangle $A'B'C'$.

A point $A$, a line $\ell$ and a circle $O$ are drawn on the plane. Construct a point $M$ on the circle $O$ such that $\ell$ divides the segment $AM$ in half.

Pentagon $ABCDE$ is inscribed in a circle. It is known that rays $AE$ and $CD$ intersect at point $P$, and rays $ED$ and $BC$ intersect at point $Q$ so that PQ || AB. Prove that $AD = BD$.

In a triangular pyramid $SABC$, every two opposite edges are equal, $O$ is the center of the circumscribed sphere. Let $A_1, B_1, C_1$ be the midpoints of the edges $BC, CA$ and $AB$, respectively. Find the radius of the circumscribed sphere of the triangular pyramid $OA_1B_1C_1$ if $BC = a, CA = b$ and $AB = c$.

In a triangle, the length of the altitude dropped to side $a$ is equal to $h_a$, and respectively to side $b$ is $h_b$. Prove that if $a> b$ then $a + h_a> b + h_b$.

Given plane $\alpha$, straight line $\ell$ in plane $\alpha$ and point $A$ outside plane $\alpha$. Consider the locus of points $M$, lying in the plane $\alpha$, such that the common perpendicular to lines $\ell$ and $AM$ passes through the midpoint of the segment $AM$. Prove that this is locus us a pair of straight lines parallel to $\ell$.

In an acute-angled triangle $ABC$, the angle $B$ is equal to $60$ degrees, $AM$ and $CN$ are its altitudes , and $Q$ is the midpoint of the side $AC$. Prove that the triangle $MNQ$ is equilateral.

Inside the convex quadrilateral $ABCD$, point $O$ is chosen so that the radii of the circles circumscribed about the triangles $AOB, BOC, COD$ and $AOD$ are equal. For which quadrangles $ABCD$ does such a point $O$ exist?

In a convex quadrilateral $ABCD$, $\angle A = \angle D$. Perpendicular bisectors of sides $AB$ and $CD$ meet at point $P$, which lies on side $AD$. Prove that the diagonals $AC$ and $BD$ are equal.

In triangle $ABC$, point $D$ is selected on side $AB$, and point $E$ on side $AC$ so that $BD = CE$. Let $F$ be a point of intersection of the circumscribed circles of triangles $ACD$ and $ABE$, different from $A$. Prove that $F$ lies on the bisector of the angle $BAC$.

Is it possible to draw thirteen straight lines on the coordinate plane so that any two straight lines intersect at a point with integer coordinates and no three pass through one point?

Point $K$ is the midpoint of side $BC$ of square $ABCD$. On the segment $AK$, a point $E$ is taken such that $CE = BC$. Find the angle $AED$ .

In an acute-angled triangle $ABC$ the altitudes $AD$ and $CE$ are drawn, $H$ is the point of their intersection. On the segments $AH$ and $CH$, points $K$ and $M$ are taken such that the $\angle  BKC = \angle AMB = 90^o$. Prove that $BM = BK$.

The line $\ell$ passes through the center $I$ of the incircle of triangle $ABC$. Lines $m_A, m_B, m_C$ are symmetric to the corresponding bisectors of the triangle wrt to $\ell$ and intersect, respectively, lines $BC, CA, AB$ at points $A', B', C'$ . Prove that $A', B', C'$ lie on the line tangent to the incircle.

Given a triangle with sides $a, b$ and $c$, which satisfy the relation $a/(b + c) = c/(a + b)$. One of the angles of this triangle is $80$ degrees. Find the rest of the angles.

Find all regular quadrangular pyramids that have a section that is a regular pentagon.

Given a quadrangular pyramid $SABCD$. Let $O$ be the intersection point of the diagonals $AC$ and $BD$. It turned out that the bases of the perpendiculars dropped from point $O$ to the side faces of the pyramid lie in the same plane. Prove that they lie on the same circle.

Points $A'$ and $B'$ lie on side $AB$ of an acute-angled triangle $ABC$. Prove that the distance from the center of the circumcircle of triangle $ABC$ to line $AB$ is less than the distance from the center of the circumscribed circle of triangle $A'B'C$ to this line.

Point $M$ is the midpoint of side $AB$ of triangle $ABC$. Point $N$ lies on the side $AC$, and $\angle ANM =\angle  BNC$. Find the ratio $MN: NB$ .

In a right-angled triangle $ABC$, the bisector of angle $A$ is perpendicular to one of the medians. What can the degree measure of angle $A$ be equal to?

Construct an isosceles triangle given the feet of the bisectors of its angles.

Given a triangle $ABC$. On the extension of side $BC$ beyond point $B$, we took a point $D$ such that $BD = BA$, point $M$ is the midpoint of side $AC$. The bisector of angle $ABC$ meets line $DM$ at point $P$. Prove that angles $BAP$ and $ACB$ are equal.

Two circles $w_1$ and $w_2$ meet at points $A$ and $B$. Circle $w_2$ passes through the center of $w_1$. The tangent to $w_2$, drawn through point $B$, intersects $w_1$ at point $C$ (different from $B$). Prove that $AB = BC$.

Given a triangle $ABC$. The inscribed circle was projected onto the straight lines containing the sides of the triangle. Prove that the six ends of the projections lie on the same circle.

In rhombus $ABCD$ on segment$ BC$ there is a point $E$ such that $AE = CD$. The segment $ED$ intersects the circumcircle of the triangle $AEB$ at point $F$. Prove that points $A, F$ and $C$ lie on one straight line.

In a triangle, each bisector is divided by the point of intersection of the bisectors in the same ratio. Which one?

A right-angled triangle is inscribed in the parabola $y = x^2$, the hypotenuse of which is parallel to the axis $Ox$. What can be the altitude of the triangle, drawn to the hypotenuse?

From the midpoints of the sides of an acute-angled triangle, perpendiculars are drawn to the adjacent sides. The resulting six straight lines bound the hexagon. Prove that its area is half the area of the original triangle.

In a convex quadrilateral $ABCD$, $\angle  A = \angle  D$. The perpendicular bisectors to the sides $AB$ and $CD$ meet at point $P$ lying on the side $AD$. Prove that the diagonals $AC$ and $BD$ are equal.

$BH$ is the altitude of an isosceles triangle $ABC$ ($AB = BC$), $M$ is the midpoint of side $AB, K$ is the second intersection point of line $BH$ with circumcircle of $BMC$. Prove that $BK = 3 / 2R$, where $R$ is the radius of the circumcircle of triangle $ABC$.

Inside the equilateral (not necessarily regular) $11$-gon $A_1A_2...A_{11}$, an arbitrary point $O$ is taken. From it, perpendiculars are dropped to the sides of the $11$-gon. Their bases $H_1, H_2, ..., H_{11}$ lie on the sides $A_1A_2, A_2A_3, ..., A_{10}A_{11}$, respectively (and not on their extensions). Prove that $A_1H_1 + A_2H_2 +...+ A_{11}H_{11} = H_1A_2 + H_2A_3 +... + H_{11}A_1$.

Pentagon $ABCDE$, in which $BC = CD$, is inscribed in a circle. Let $P$ be the intersection point of the diagonals $AC$ and $BE, Q$ the intersection point of the diagonals $AD$ and $CE$. Prove that lines $PQ$ and $BD$ are parallel.

All angles of pentagon $ABCDE$ are equal. Prove that the perpendicular bisectors of segments $AB$ and $CD$ meet at the bisector of angle $E$.

Given a triangle $ABC$. Three straight lines are drawn through the point $P$, perpendicular to the straight lines $AC, BC$ and the line containing the median $CE$, respectively. Let these lines intersect the altitude $CD$ at points $K, L, M$, respectively. Prove that $KM = LM$.

Six points are located on a plane so any three of them serve as the vertices of a triangle with sides of different lengths. Prove that the smallest side of one of the triangles is also the largest side of the other triangle.

In triangle $ABC$ altitudes $AA_1$ and $BB_1$ are drawn. The circumscribed circles of triangles $ABC$ and $A_1B_1C$ are tangent. Prove that triangle $ABC$ is isosceles.

The billiard table has the shape of a right-angled triangle. From the point of the hypotenuse, a ball was released perpendicular to it, which hit two sides and returned to the hypotenuse. Prove that the length of such a path does not depend on the starting point.

(The ball is reflected from the sides according to the law "the angle of incidence is equal to the angle of reflection".)

The quadrilateral$ ABCD$ is inscribed in the circle. Lines $AB$ and $CD$ meet at point E, lines $AD$ and $BC$ at point $F$. The bisector of angle $AEC$ intersects side $BC$ at point $M$ and side $AD$ at point $N$, and the bisector of angle $BFD$ intersects side $AB$ at point $P$ and side $CD$ at point $Q$. Prove that the quadrilateral $MPNQ$ is a rhombus.

In triangle $ABC$, side $AB$ is equal to the half-sum of the other two. Prove that the angle $OIC$ is a right angle .($O, I$ are the circumcenter and the incenter of $ABC$ respectively).

The convex polyhedron $W$ has the following properties:
a) it has a center of symmetry;
b) the section of the polyhedron $W$ by the plane passing through the center of symmetry and any edge has the form of a quadrangle;
c) there is a vertex of the polyhedron $W$ that belongs to exactly three edges.
Prove that $W$ is a parallelepiped.

Given circle $O_1$, point $A$ inside and point $B$ outside it. Construct a circle passing through $A$ and $B$and at the intersection with $O_1$ giving a chord of the smallest length.

What is the minimum width of an infinite strip from which any triangle of area $S$ can be cut?

All vertices and centers of all faces were marked on the cube ($14$ points in total). It turned out that the distance from any of these points to a certain plane takes only two different values, the smaller of which is equal to $1$. Find the length of the edge of the cube.

From the point $M$ of the side $AC$ of an equilateral triangle $ABC$, the perpendiculars $MX$ and $MY$ are dropped to the sides $AB$ and $BC$, respectively. Point $O$ is the center of triangle $ABC$. Prove that line $OM$ divides the segment $XY$ in half.

Median $AM$, angle bisector $BL$, and altitude $CH$ all intersect at one point. Is a triangle necessarily equilateral?

Quadrilateral $ABCD$ has no parallel sides. Points $E$ and $F$ are such that $EBCD$ and $ABFD$ are parallelograms. Prove that if points $A, C, E, F$ do not lie on one straight line, then $AEFC$ is a parallelogram.

$ABC$ is an acute-angled triangle. On the side $AB$, as on the diameter, a circle was built that intersects the altitude $CC'$ and its extension at points $M$ and $N$. The circle constructed on the diameter $AC$ intersects the altitude $BB'$ and its extension at the points $P$ and $Q$. Prove that the points $M , N, P, Q$ lie on the same circle.

In quadrilateral $ABCD$, angles $A$ and $C$ are right angles, and $ADB$ is twice the angle $BDC$. Point $K$ is such that point $C$ is the midpoint of segment $BK$, and $O$ is the intersection point of the diagonals of the quadrilateral $ABCD$. Find the angle $KOD$.

A rhombus is circumscribed around a square with side $7$, and the diagonals of the rhombus are parallel to the sides of the square. Find its diagonals if it is known that they are equal to integers.

The diagonal $AC$ of the inscribed quadrilateral $ABCD$ is divided by points $P$ and $Q$ into $3$ equal parts so that $P$ lies between $A$ and $Q$. Lines $BP$ and $DQ$ meet at point $R$, and $RA = RC$. Prove that if points $L$ and $M$ divide the diagonal $BD$ into three equal parts so that $L$ lies between $B$ and $M$, then the intersection point of lines $AL$ and $CM$ is equidistant from $B$ and $D$.

$A_1A_2A_3A_4A_5A_6A_7A_8A_9$ is a regular nonagon. Which is greater - the sum of the areas of triangles $A_1A_2A_9$, $A_3A_8A_9$, $A_4A_7A_8$, and $A_5A_6A_7$, or the sum of the areas of triangles $A_2A_3A_9$, $A_3A_4A_8$, and $A_4A_5A_7$?

The bisector of angle $A$ of an acute-angled triangle $ABC$ intersects the circle circumscribed around the triangle at point $Z$. The perpendicular bisectors of sides $AB$ and $AC$ intersect $AZ$ at points $X$ and $Y$. Prove that $AX = ZY$.

A point is selected on each edge of the tetrahedron. Prove that the volume of at least one of the four resulting tetrahedra (adjacent to the vertices of the original) does not exceed $1/8$ of the volume of the original tetrahedron.

Prove that in any triangle, the ratio of the smallest altiude to the smallest bisector is greater than $2^{-1/2}$.

In the triangular pyramid $SABC$, the edges satisfy the equalities $AB^2 + CS^2 = AC^2 + BS^2 = AS^2 + BC^2$. Prove that at least one of the pyramid's faces is an acute-angled triangle.

Construct a square $ABCD$ if the points $E, F, G, H$ are marked such that $E$ lies on $AB, F$ lies on $BC, G$ lies on $CD, H$ lies on $DA$, and it is known that the solution is unique.

In quadrilateral $ABCD$, the sum of angles $ABD$ and $BDC$ is $180^o$, and sides $AD$ and $BC$ are equal. Prove that the angles at the vertices $A$ and $C$ of such a quadrangle are equal.

Find the angles of an isosceles triangle with the centers of the inscribed and circumscribed circle symmetric wrt the base.

The vertices of a convex quadrilateral lie inside a regular triangle with side $1$. Prove that the length of at least one side of the quadrilateral is less than $0.5$.

Let $O$ be the center of a circle $w$ circumscribed around an acute-angled triangle $ABC, W$ be the midpoint of that arc $BC$ of circle $w$ that does not contain point $A$ and $H$ be the point of intersection of the altitudes of triangle $ABC$. Find the angle $BAC$ if $WO = WH$.

Points $K$ and $L$ are projections of vertices $A$ and $C$ of an acute-angled triangle $ABC$ onto the bisector of the outer angle at vertex $B$. Points $H$ and $M$ are the feet of the altitude and median drawn from vertex $B$. Prove that points $H, M, K$ and $L$ lie on the same circle.

Given a triangle $ABC$. Two straight lines symmetric to the straight line $AC$ wrt the straight lines $AB$ and $BC$, respectively, intersect at the point $K$. Prove that the straight line $BK$ passes through the center of the circumscribed circle of the triangle $ABC$.

Let $A, B, C$ be angles, $a, b, c$ sides of a triangle. Prove the inequality$$60^o \le \frac{aA+bB+cC}{a+b+c}\le 90^o$$

Is there a square whose vertices lie on four concentric circles whose radii form an arithmetic progression?

On the side $AC$ of $ABC$, point $K$ is chosen, and on the median $BD$, point $P$ so that the area of triangle $APK$ is equal to the area of triangle $BPC$. Find the locus of the points of intersection of lines $AP$ and $BK$.

Three straight lines are drawn parallel to the sides of the triangle. Each of the straight lines is removed from the side to which it is parallel by a distance equal to the length of this side. Moreover, for each side of the triangle, a straight line parallel to it and a vertex opposite to this side are located on opposite sides of it. Prove that the intersection points of the extensions of the sides of a triangle with three drawn lines lie on the same circle.

Is there a regular hexagon whose vertices lie on six concentric spheres whose radii form an arithmetic progression?

From the medians of a right-angled triangle, you can make another right-angled triangle. Prove that these triangles are similar.

Points $A_1, B_1, C_1, D_1$ are taken on the lateral edges $SA, SB, SC$ and $SD$ of the regular quadrangular pyramid $SABCD$. Prove that they lie in the same plane if and only if the distances $a, b, c, d$ from them to the vertex $S$, respectively, satisfy the relation$$\frac{1}{a}+\frac{1}{c}=\frac{1}{b}+\frac{1}{d}.$$

In a right-angled triangle $ABC$, the angle is $C = 90^o$. $K$ is the midpoint of $AB$, on the side of $BC$ we took $N$ such that $2CN=NB$. Find the ratio $AN: KN$.

Given a triangle $ABC$. The perpendiculars drawn on sides $AB$ and $AC$ at points $B$ and $C$ intersect at point $A_1$, the perpendiculars drawn on sides $BA$ and $BC$ at points $A$ and $C$, intersect at point $B_1$, the perpendiculars drawn on sides $CA$ and $CB$ at points $A$ and $B$ intersect at point $C_1$. Prove that triangles $ABC$ and $A_1B_1C_1$ are congruent.

Two squares are drawn on the plane - $ABCD$ and $KLMN$ (their vertices are listed counterclockwise). Prove that the midpoints of the segments $AK, BL, CM, DN$ are also the vertices of the square.

A segment $AB$ and a point $C$ are given on the plane. Find the set of points $M$ such that $r (M, C) = r (M, AB)$, where $r (M, C)$ is the distance between $M$ and $C$, and $r (M, AB)$ is the shortest distance from $M$ to $AB$.

An arbitrary regular $2n$-gon $P '$ with side $1$ is placed inside a regular $2n$-gon $P$ with center $O$ and side $2$. Prove that $P'$ covers $O$.

You are given a circle tangent to its line $\ell$ and point M on $\ell$. Find the locus of the vertices $C$ of triangles $ABC$ circumscribed about a given circle, whose base $AB$ lies on $\ell$, and $CM$ is the median.

The bisector $AL$, the altitude $BH$ and the median $CM$ are drawn in the triangle $ABC$. It turned out that the angles $CAL, ABH$ and $BCM$ are equal. What angles could triangle $ABC$ have?

Given a triangle $ABC$. Find inside it a point $O$ with the following property: for any straight line passing through point $O$ and intersecting the sides of triangle $AB$ at point $K$ and $BC$ at point $L$, holds the equality$$\frac{AK}{KB}+ \frac{CL}{LB}=1$$

You are given a circle and a point $P$ inside it. Find the locus of the vertices $D$ of isohedral tetrahedra $ABCD$, whose base $ABC$ is inscribed in this circle and has a center of gravity at point $P$.

Is it possible to place a cube in some right circular cone so that the seven vertices of the cube lie on the lateral surface of the cone?

Prove that the product of the distances from the center of the inscribed circle to the vertices of the triangle is equal to the product of the square of the diameter of the inscribed circle and the radius of the circumscribed circle.

You are given a circle $O$, a point $A$ lying on it, a perpendicular to the plane of circle $O$, drawn from point $A$, and a point $B$ lying on this perpendicular. Find the locus of the bases of the perpendiculars dropped from point $A$ onto the straight lines passing through point $B$ and an arbitrary point of the circle $O$.

In triangle $ABC$, point $M$ lies on side $AC$, points $C_1, B_1, M_1, M_2$ are the midpoints of segments $AB, AC, MC, MB$, respectively, $C_1M_1 = B_1M_2$. Prove that triangle $ABC$ is right-angled.

The distance between two parallel lines $l$ and $m$ is $1$. The side of square $ABCD$ is also $1$. Vertices $A$ and $C$ lie in the strip between $l$ and $m$, and vertices $B$ and $D$ are outside the strip. Sides $AB$ and $BC$ meet $l$ at points $X$ and $Y$, and sides $CD$ and $AD$ meet $m$ at points $Z$ and $T$, respectively. Prove that the angle between lines $XZ$ and $YT$ is $45^o$.

Oral Moscow Team MO 2007 8B p2  (there is a typo)
$ABCD$ is a parallelogram. Find its angles if it is known that $AD = 2DB$ and $\angle ABD$ is three times larger than $\angle  DBC$.

In triangle $ABC$, vertices $A$ and $C$, the center of the incircle and the center of the circumcircle lie on the same circle. Find $\angle B$.

Points $A$ and $B$ are given on the plane. Find the locus of the vertices $C$ of acute-angled triangles $ABC$, for which the altitude drawn from the vertex $B$ is equal to the median drawn from the vertex $A$.

Points $K$ and $L$ were selected on sides $BC$ and $CD$ of square $ABCD$, respectively. $P_1$ and $P_2$ are the bases of the perpendiculars dropped on lines $AK$ and $AL$ from point $B, Q_1$ and $Q_2$ are the bases of perpendiculars dropped on lines $AK$ and $AL$ from point $D$. Prove that the segments $P_1P_2$ and $Q_1Q_2$ are equal and perpendicular.

Two intersecting chords $AB$ and $CD$ are drawn in the circle. A point $M$ is taken on the chord $AB$ so that $AM = AC$, and on the chord $CD$ there is a point $N$ such that $DN = DB$. Prove that if the points $M$ and $N$ do not coincide, then the line $MN$ is parallel to the line $AD$.

In triangle $ABC, I$ is the incenter. Let $N, M$ be the midpoints of sides $AB$ and $CA$, respectively. Lines $BI$ and $CI$ meet $MN$ at points $K$ and $L$, respectively. Prove that $AI + BI + CI> BC + KL$.

Similar isosceles triangles $QPA$ and $SPB$ are drawn outside the parallelogram $PQRS$ (where $PQ = AQ$ and $PS = BS$). Prove that $RAB, QPA$ and $SPB$ are similar.

Determine the type of quadrangle $ABCD$ of area $S$ if there is a point $O$ inside it, for which the equality $2S = OA^2 +OB^2 + OC^2 +OD^2$ is fulfilled.

The altitudes of the triangle $ABC$ intersect at point $O$, and points $A_1, B_1, C_1$ are the midpoints of the sides $BC, CA, AB$, respectively. A circle centered at $O$ intersects line $B_1C_1$ at points $D_1, D_2$, line $C_1A_1$ at points $E_1, E_2$, and line $A_1B_1$ at points $F_1, F_2$. Prove that $AD_1 = AD_2 = BE_1 = BE_2 = CF_1 = CF_2$.

In a tetrahedron $ABCD$, the angle $BAC$ is equal to the angle $ACD$ , and the angle $ABD$ is equal to the angle $BDC$ . Prove that $AB = CD$.

You are given a trihedral angle with apex $O$, all of whose flat angles are right. Points $A, B$ and $C$ are selected on its edges, one on each. Find the locus of the points of intersection of the bisectors of all possible triangles $ABC$.

From point $P$ inside an acute-angled triangle $ABC$, perpendiculars $PA_1, PB_1$ and $PC_1$ are dropped to the sides $BC, CA$ and $AB$, respectively. Determine the position of point $P$, if it is known that all three quadrangles $PA_1BC_1$, $PB_1CA_1$ and $PC_1AB_1$ are tangential.

Circles $S_1$ and $S_2 $ with centers $O_1$ and $O_2$ intersect at points $A$ and $B$. Circle, passing through points $O_1,O_2$ and $A$, again intersects circle $S_1$ at point $D$, circle $S_2$ at point $E$ and line $AB$ at point $C$. Prove that $CD = CB = CE$.

Two circles of radius $R$ and $r$ touch the straight line $\ell$ at points $A$ and $B$ and intersect at points $C$ and $D$ ($D$ lies closer to $I$ than $C$). Prove that the radius of the circle circumscribed around the triangle $ABC$ does not depend on the length of the segment $AB$.

Given a quadrangle $ABCD$, in which $AB = AD$ and $\angle ABC = \angle ADC  = 90^o$. On the sides $BC$ and $CD$, points $F$ and $E$ are selected, respectively, so that $DF \perp AE$. Prove that $AF \perp BE$.

In an acute-angled triangle $ABC$, with the altitude $BK$ as on the diameter a circle $S$ is constructed, intersecting the sides $AB$ and $BC$ at points $E$ and $F$, respectively. Tangents are drawn to circle $S$ at points $E$ and $F$. Prove that their intersection point lies on the median of the triangle from vertex $B$.

Let's call a line connecting the midpoints of crossing edges of a tetrahedron a good middle line of a tetrahedron if it makes equal angles with four straight lines containing the remaining edges of the tetrahedron. Prove that a tetrahedron is regular if at least two of its middle lines are good.

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ intersect at points $A$ and $B$. Ray $O_1B$ intersects $S_2$ at point $F$, and ray $O_2B$ intersects $S_1$ at point $E$. A straight line passing through point $B$ parallel to line $EF$, intersects for the second time circles $S_1$ and $S_2$ at points $M$ and $N$, respectively. Prove that $MN = AE + AF$

Each vertex of the trapezoid was reflected symmetrically with respect to a diagonal not containing this vertex. Prove that if the resulting points form a quadrilateral, then it is also a trapezoid.

In triangle $ABC$ ($AB <BC$), point $I$ is the center of the inscribed circle, $M$ is the midpoint of the side $AC , N$ is the midpoint of the arc $ABC$ of the circumscribed circle. Prove that $\angle IMA = \angle INB$.

The cosines of the angles of one triangle are respectively equal to the sines of the angles of the other triangle. Find the largest of the six angles of these triangles.

Given a triangle $ABC$, in which $AB = BC, AB \ne AC$. On the side $AB$, point $E$ is selected, on the extension of AC beyond point $A$, point $D$ is selected so that $\angle BDC = \angle ECA$. Prove that the areas of triangles $BEC$ and $ABC$ are equal.

Is there a triangular pyramid, each base edge of which is seen from the middpoint of an opposing side edge at a right angle?

Each vertex of the convex quadrangle of area $S$ is reflected symmetrically with respect to the diagonal not containing this vertex. Let's denote the area of the resulting quadrangle by $S'$. Prove that $\frac{S'}{S} <3$.

Let $AA_1$and $BB_1$ be the altitudes of an acute-angled non-isosceles triangle $ABC$. It is known that the segment $A_1B_1$ intersects the midline parallel to $AB$ at point $C'$. Prove that the segment $CC'$ is perpendicular to the straight line passing through the intersection of the altitudes and the center of the circumscribed circle of the triangle $ABC$.

Prove that any non-isosceles triangle has two sides whose ratio is greater than $3/5$ but less than $1$.
In the convex quadrilateral $ABCD$, point $E$ is the midpoint of side $AD$, and point $F$ is the midpoint of side $BC$. The segments $BF$ and $CE$ intersect at point $O$. Rays $AO$ and $BO$ divide the side $CD$ into three equal parts. Find the ratio of $AB$ to $BC$.

What is the largest $k$ for the statement:
“in any triangle there are two sides, the ratio of which is greater than $k$, but less than $1 / k$”?

The bisectors of the internal angles $A, B, C$ of the triangle $ABC$ intersect the circumscribed circle, respectively, at points $A', B', C'$. Point $I$ is the center of the inscribed circle $ABC$. The circle with a diameter $A'I$ intersects the side $BC$ at points $A_1, A_2$. Points $B_1, B_2, C_1, C_2$ are defined similarly. Prove that all points $A_1,A_2, B_1, B_2, C_1, C_2$ lie on the same circle.

Inside the triangle $ABC$, in which $\angle A = 60^o$, a point $T$ is chosen such that $\angle ATB =\angle ATC = 120^o$. Let $M, N$ be the midpoints of sides $AB, AC,$ respectively. Prove that points $A, M,T, N$ lie on the same circle.

An arbitrary point $M$ is taken on the larger arc $AB$. From the midpoint$ K$ of the segment $MB$, a perpendicular is dropped onto the line $MA$. Prove that all such perpendiculars pass through one point, independent of the position of point $M$.

Let$ B_0$ be the midpoint of side $AC$ of triangle $ABC$. Drop from the midpoint of the segment $AB_0$ perpendicular on the side BC, and from the midpoint of the segment $B_0C$ perpendicular on the side $AB$ . Let's denote the point of intersection of these perpendiculars by $B'$. Similarly, construct points $C'$ and $A'$. Prove that triangles $A'B'C'$ and $ABC$ are similar.

Diagonals of an inscribed quadrilateral $ABCD$ intersect at point $E$ (see fig.). Let $I_1$ be the center of a circle inscribed in triangle $ABC$, and $I_2$ be the center of the circle, inscribed in a triangle $ABD$. Prove that line $I_1I_2$ cuts off an isosceles triangle from triangle $AEB$.

Prove that the four line segments connecting the vertices of the tetrahedron with the centers circles inscribed in opposite faces intersect at one point if and only if three products of the lengths of opposite edges are equal to each other.

Segments $AA_1$ and $BB_1$ are bisectors of triangle ABC. Prove that the segment $A_1B_1$ intersects the inscribed circle of triangle $ABC$.

Let $O$ be the intersection point of the diagonals of the parallelogram $ABCD, P$ be the second point intersection of a circle passing through points $A, O, B$ with line $BC$. Prove that line $AP$ touches the circle passing through points $A, O, D$.
    
Let $P$ be an arbitrary point inside the triangle $ABC$. Let's choose some vertex and reflect it wrt $P$, and then reflect the resulting point wrt the midpoint of the side opposite the selected vertex. We denote the resulting point by $Q$. Prove that $Q$ does not depend on the choice of the vertex of the triangle $ABC$.

In triangle $ABC$, side $AB$ is larger than $AC$. On the extension of side $AB$ beyond point $A$ is chosen point $P$ such that $AP + PC = AB$. Segment $AM$ is the median of triangle  $ABC$, point $Q$ on $AB$ is such that $CQ\perp AM$. Prove that $BQ = 2AP$.

The bisector of angle $A$ of triangle $ABC$ intersects its circumcircle at point $D$. It so happened that $2AD^2= AB^2 + AC^2$. Find the angle between straight lines $AD$ and $BC$

Perpendiculars $AB_1, AC_1, AD_1$ were dropped in the tetrahedron $ABCD$ from the vertex $A$ on the plane dividing the dihedral angles at the edges $CD, BD, BC$ in half. Prove that the plane $(B_1C_1D_1)$ is parallel to the plane $(BCD)$.

Let $AA_1$ and $BB_1$ be the altitudes of an acute-angled non-isosceles triangle $ABC$. It is known that the segment $A_1B_1$ intersects the midline parallel to $AB$ at point $C'$. Prove that the segment $CC'$ is perpendicular to the straight line passing through the point of intersection of the altitudes and the center of the circumcircle of triangle $ABC$.

In an acute-angled triangle $ABC$ through the center $O$ of the circumscribed circle and the vertices $B$ and $C$ a circle is drawn. Let $OK$ be the diameter of this circle, and $D$ and $E$ be the points of its intersection with lines $AB$ and $AC$, respectively. Prove that $ADKE$ is a parallelogram.

Given a circle $\omega$, a point $A$ inside this circle, and a point $B$ different from $A$. All possible triangles $BXY$ are considered such that the segment $XY$ is a chord of circle $\omega$ passing through point $A$. Prove that the centers of the circles circumscribed around these triangles lie on one straight line.

In an acute-angled triangle$ ABC$ ($AC> AB$) , $M$ is the midpoint of side $BC, D$ is the projection of the point of intersection of the altitudes of the triangle on the tangent to the circumscribed circle of the triangle at the vertex $A$. Prove that triangle $AMD$ is isosceles.

Is it possible to choose three points $A, B$ and $C$ on the plane so that for any point $P$ the length of at least one of the segments $AP, BP$ and $CP$ would be an irrational number?

A convex quadrilateral ABCD is drawn on the board. Hooligan Vasya, going up to the board, changes its vertex $B$ to a point symmetric to $B$ wrt the perpendicular on $AC$ (the old vertex erases, the new one is designated by $B$). Hooligan Petya, approaching the board, changes vertex $C$ to a point, symmetric to $C$ wrt the perpendicular on $BD$. They walked up to the board in this order: Vasya, Petya, Vasya, Petya, Vasya, Petya (exactly six times). It is known that after each the hooliganism the quadrilateral remained convex. Prove that at the end you got the original quadrangle.

A regular tetrahedron $ABCD$ is given. All possible points $M$ are considered on the edge $CD$. Prove that the orthocenters of all $AMB$ triangles lie on the same circle.

Point $O$ is the center of the circumscribed circle of an acute-angled triangle $ABC$. On the ray $BO$ lies point $P$. The circumscribed circles of triangles $CPB$ and $APB$ intersect sides $AB$ and $BC$ at points $K$ and $L$, respectively. Prove that the midpoint of the segment $KL$ is equidistant from vertices $A$ and $C$.

In a triangle $ABC$ inscribed in a circle, $AB <AC$. On the side $AC$ , point $D$ is marked so that $AD = AB$. Prove that the perpendicular bisector of the segment $DC$ bisects the arc $BC$ that does not contain point $A$.

Prove that you can always compose a trapezoid or parallelogram from segments that are sides of any convex quadrilateral.

You are given a convex polygon with an odd number of sides. The lengths of all its sides are equal to $1$. Prove that its area is at least $\sqrt3 /4$
 
Two disjoint circles $\omega_1$ and $\omega_2$ and points $A$ and $B$ on them so that the segment $AB$ has the greatest possible length. Let $AR$ and $AS$ be tangents drawn from point $A$ to circle $\omega_2$, $BP$ and $BQ$ be  tangents from point $B$ to the circle $\omega_1$. The circle $\gamma_1$ touches the circle $\omega_1$ internally and the rays $AR$ and $AS$. The circle $\gamma_2$ touches circle $\omega_2$ internally and the rays $BP$ and $BQ$. Prove that the radii of the circles $\gamma_1$ and  $\gamma_2$ are equal.
 
The tetrahedron $ABCD$ satisfies the equality $\angle BAC + \angle BAD = \angle ABC + \angle ABD = 90^o$. Let $O$ be the center of the circumscribed circle of triangle $ABC$, $M$ be the midpoint of the edge $CD$. Prove that lines $AB$ and $MO$ are perpendicular.

In a non-isosceles acute-angled triangle $ABC$, the altitudes $AA_1$ and $CC_1$ are drawn, $H$ is the point of intersection of altitudes, $O$ is the center of the circumscribed circle, $B_0$ is the midpoint of side $AC$. Line $BO$ intersects the side $AC$ at $P$, lines $BH$ and $A_1C_1$ intesect at $Q$. Prove that lines $HB_0$ and $PQ$ are parallel. 

Oral Moscow Team MO 2015 IX p5 (2000 Argentina OMA L2 p5)
In parallelogram $ABCD$, points $M$ and $N$ are the midpoints of sides $BC$ and $CD$, respectively. Can the rays $AM$ and $AN$ divide the angle $BAD$ into three equal parts?

The quadrilateral $ABCD$ is both cyclic and tangential, and the inscribed in $ABCD$  circle touches its sides$ AB, BC, CD$ and $AD$ at points $K, L, M, N$, respectively. The bisectors of the external angles $A$ and $B$ of the quadrilateral intersect at the point $K '$, bisectors of the external angles $B$ and $C$ intersect  at point $L '$, bisectors of the external angles $C$ and $D$ intersect  at point $M'$, bisectors of the external angles $D$ and $A$ intersect  at point $N '$. Prove that lines $KK ', LL', MM '$ and $NN'$ pass through one point.

Let $P$ be  an arbitrary point on the side $AC$ of triangle $ABC$. On the sides $AB$and$ BC$, points $M$ and $N$ are taken respectively so that $AM = AP$ and $CN = CP$.  The perpendiculars drawn from points $M$ and $N$ on the sides $AB$ and $BC$ respectively, intersect at the point $Q$. Prove that the angle $\angle QIB = 90^o$  where $I$ is the center of the circle inscribed in the triangle $ABC$.

In an acute-angled triangle $ABC$ point $M$ is midpoint of the side $BC$,  points $D$ and $E$ are projections of point $M$ on the sides $AB$ and $AC$ respectively. The circles circumscribed around triangles $ABE$ and $ACD$ , intersect at the point $K$ different from $A$.  Prove that lines $AK$ and $BC$ are perpendicular.

Point $I$ is the center of incircle of triangle $ABC$ and point $M$ is the midpoint of side $BC$. Line $MI$ intersects side $AB$ at point $D$, and line through $B$ perpendicular to $AI$ intersects the segment $CI$ at point $K$. Prove that $KD$ is parallel to $AC$.

Alexey has a device that measures the distance from all the vertices of the cube and intersection points of the diagonals of the faces to some selected plane and shows them on screen. Once Alexey saw that on the screen there are only two different numbers among all the numbers. The smallest is $2$. What is the edge of a cube?

Three points are given that do not lie on one straight line. Construct a circle centered on one of the them, such that the tangents drawn to it from the other two points would be parallel.

The radius of the circumscribed circle of triangle $ABC$ is equal to the radius of the circle tangent to the side $AB$ at the point $C$ and the extensions of the other two sides at the points A' and $B'$. Prove that the center of the circumscribed circle of triangle $ABC$ coincides with the orthocenter of triangle $A'B'C'$.

A triangle$ ABC$ is given on the plane. Find the interior point of the triangle, the product of the distances from which to its sides is maximum.

The pentagon $ABCDE$ is inscribed in the circle, the side of which is $BC =\sqrt{10}$ . The diagonals $EC$ and $AC$ meet the diagonal $BD$, respectively, at points $L$ and $K$. It turned out that that a circle can be circumscribed around the quadrilateral $AKLE$. Find the length of the tangent segment  drawn from point $C$ to this circle.

All faces of the tetrahedron $ABCD$ are acute-angled triangles. The altitudes $AK$ and $AL$ were drawn at the edges $ABC$ and $ABD$, respectively. It turned out that points $C, K, L$ and $D$ lie on the same circle. Prove that $AB$ is  perpendicular to $CD$.

Given an acute-angled triangle $ABC$, where$ P, M, N$ are the midpoints of sides $AB, BC, AC$ respectively. From some point $H$ inside the triangle, the perpendiculars $HK, HS$ and $HQ$ were dropped to sides $AB, BC, AC$ respectively. It turned out that $MK = MQ, NS = NK, PS = PQ$. Prove that H is a point intersection of altitudes of triangle $ABC$.

On the diagonal $AC$ of the rhombus $ABCD$, a point $E$ is taken, which is different from points $A$ and $C$, and on the lines $AB$ and $BC$ are points $N$ and $M$, respectively, with $AE = NE$ and $CE = ME$. Let $K$ be the intersection point of lines $AM$ and $CN$. Prove that points $K, E$ and $D$ are collinear.

Is it possible to place four equal polygons on a plane so that each two of them have no common interior points, but have a common border segment?

In the parallelogram $ABCD$, points $M$ and $N$ are selected on the sides $AB$ and $BC$, respectively, where $AM = CN$, $Q$ is the intersection point of the segments $AN$ and $CM$. Prove that $DQ$ is the bisector of angle $D$.

Snow White has a piece of velvet in the form of a square with a side of $27.5$ cm, on which contains four spots. Is it guaranteed to be cut from of this square is a pure square piece with a side of $10$ cm? (We count the spots as points.)

A circle inscribed in an angle with a vertex $O$ touches its sides at points $A$ and $B, K$ is an arbitrary point on the smaller of the two arcs $AB$ of this circle. On line $OB$, a point $L$ is taken such that lines $OA$ and $KL$ are parallel. Let $M$ be the point of intersection of the circle $\omega$, circumscribed around the triangle $KLB$, with the line $AK$, different from $K$. Prove that line $OM$ is tangent to the circle $\omega$.

Each segment of the spatial closed polyline $ABCD$ touches a sphere. Prove that all tangency points lie in the same plane.

In a right triangle $ABC$ ($\angle A = 90^o$) on the side $AC$, take point $D$.  Point $E$ is symmetric to point $A$ wrt $BD$ and line $CE$ intersects the perpendicular from the point $D$ on the $CB$ , at the point $F$. Prove that lines $AF, DE, CB$ intersect at one point.

Let be $l_a, l_b, l_c$ be the lengths of bisectors of $ABC$ respective to sides $a, b, c$, and $R$ the  radius of  the circumscribed circle. Prove the inequality $$\frac{b^2+c^2}{l_a}+\frac{c^2+c^2}{l_b}+\frac{a^2+b^2}{l_c}>4R$$

In triangle $ABC$ ($AB \ne AC$), the inscribed circle centered at point  $I$ touches the side $BC$ at point $D$. Let $M$ be the midpoint of $BC$. Prove that the perpendiculars dropped from the points $M$ and $D$ on the lines $AI$ and $MI$, respectively, intersect at the altitude of the triangle $ABC$ drawn from the vertex $A$ (or on its extension).

Oral Moscow Team MO 2018 X-XI p5 (also Czech And Slovak MO, IIIA 1994 p2)
A cube contains a convex polyhedron whose projection onto any of the cube's faces covers the entire face. Show that the volume of the poiyhedron is not less than $1/3$ that of the cube. 

On the bisector of angle $A$ of triangle $ABC$ ($\angle A=30^o, \angle B=105^o$), a point $P$ is marked such that it lies inside the triangle and $PC= BC$. Find the angle $\angle APC$.

Point $L$ is chosen on the side $AC$ of triangle $ABC$. It is known that $\angle ABL = 7^o, \angle CBL = 70^o$. Prove that $AC + 2AL> BC$.

Consider a triangle $ABC$ in which a circle $\omega$ with center $I$ is inscribed, touching the side $BC$ at point $D$. Let $\omega_b$ and $\omega_c$ be the inscribed circles of triangles $ABD$ and $ACD$, respectively, touching the side $BC$ at points $E$ and $F$, respectively. Let P be the point of intersection of the segment $AD$ with the line of centers of the circles $\omega_b$ and $\omega_c$. Let $X$ be the point of intersection of the lines $BI$ and $CP$. Let $Y$ be the point of intersection of lines $CI$ and $BP$. Prove that lines $EX$ and $FY$ intersect on the inscribed circle of triangle $ABC$.

In triangle $ABC$, altitudes $BE$ and $CF$ are drawn, intersecting at point $H$. Point $A'$ is symmetric to $A$ wrt line $BC$. The circumscribed circles of triangles $AA'E$ and $AA'F$ intersect the circumscribed circle of triangle $ABC$ at points $P$ and $Q$, different from $A$. The lines $PQ$ and $BC$ intersect at the point $R$. Prove that the lines $EF$ and $RH$ are parallel.

Given $DA'+ DC'=DB'$, prove that $ABCD$ is cyclic.
(L.A. Popov)
A line $\ell$ is drawn through the vertex $D$ of the square $ABCD$  which does not intersect the sides of the square. Ray $CA$ intersects $\ell$ at point $E$. Point $F$ is selected on ray $ED$ beyond point $D$ such that  $\angle DCF =45^o$.  It turned out that $AE=CF$. Find the angle $CFD$.

(L.A. Popov)
Point $D$ is chosen on side $BC$ of triangle $ABC$. Points $E$ and $F$ are selected on sides $AC$ and $AB$, respectively, so that $BF =CD$ and $CE =BD$. Circumscribed circles of triangles $BDF$ and$ CDE$ meet for the second time at point $P$. Prove that exists point $Q$ is such that regardless of the position of point $D$, distance $PQ$ is constant.

In the triangle $ABC$ the center of the incircle $I$ and the points $B_1$ and $C_1$ touchpoints  of the excircles opposite to vertices $B$ and $C$ respectively.  Point $D$ is chosen on the line $BC$ so that $\angle AID = 90^o$. Prove that the line $AD$ is tangent to the the circumscribed to the circle of the triangle $AB_1C_1$.

In one of the cells of the square $2021 \times 2021$ there is an invisible cockroach. Lyosha has a slipper with which he hits a $50 \times 50$ square once a minute. Every time hitting the cockroach runs over to the cell adjacent to the side. Prove that Lyosha can kill the cockroach.

In a quadrilateral $ABCD$, $\angle A= \angle C=90^o$. Points $M,N$ lie on the diagonal $BD$ such that $AN  \parallel BC$ and $CM \parallel AB$. On the sides AD and CD, lie the points X and Y respectively such that $\angle XNB= \angle YMD=90^o$. Prove that the segment $AC$ is equal to the semiperimeter of triangle $BXY$.

In an acute-angled triangle $ABC$, the point $O$ is the center of the circumscribed circle. Points $E$ and $F$ are chosen on the sides $AB$ and $AC$, respectively such that $\angle BAC = \angle EOF$. Prove that the perimeter of triangle $AEF$ is not less than $BC$ .

In the tangential quadrilateral $ABCD$, the point $I$ is the center of the inscribed circle. Rays $AB$ and $DC$ meet at $F$, and rays $AD$ and $BC$ meet at $G$ . Ellipse $\omega$ with focuses at $F$ and $G$ passes through points$ B$ and $D$. The branch of the hyperbola $\gamma$ with focuses at $F$ and $G$ passes through the points $A$ and $C$. Curves $\omega$ and $\gamma$ meet at points $P$ and $Q$. Prove that points $I,P$ and $Q$ are collinear.



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