Ukraine 1991 - 2021 VIII-XI 223p

geometry problems from Ukrainian Mathematical Olympiads
with aops links in the names

collected inside aops : here 

grade VIII, grade IX, grades X-XI plane, 3D geometry 

1991 - 2021

1991 Ukraine MO grade VIII P2

On the sides $AB$ and $AC$ of the triangle $ABC$ were chosen respectively points $M$ and $N$ such that each of them divides the corresponding side in the ratio $1: 1991$ (counting from the vertex $A$). In what ratio does the intersection point of the segments $CM$ and $BN$ divide each of these segments?

1991 Ukraine MO grade VIII P5

Two lines are drawn through the ends of the diameter $AB$ of the circle, a tangent line $\ell$ is drawn through point $A$, and a secant line $m$ is drawn through point $B$. Let $P$ be the second intersection point of line m with the circle. Draw through the point $P$ tangent to it and denote by $M$ , $N$ the points of intersection of the line $\ell$ with lines $m, n$ respectively. Prove that the triangle $MNP$ is isosceles.

1991 Ukraine MO grade IX P3

On the sides $AB, BC, CA$ of the triangle $ABC$ are taken points E. F, G respctively so that $AE/AB = x$. $BF / BC = y$, $CG / CA = z$. The area of triangle $ABC$ is equal to $S$. Calculate the area of triangle $EFG$.

1991 Ukraine MO grade IX P4

A point $O$ and two congruent squares $F_1$ and $F_2$ are given on the plane of area $S$. Figure $F$ is formed by all such points $B$ that the vector $\overrightarrow{OB}$ can be written as $\overrightarrow{OB }= \overrightarrow{OA_1} + \overrightarrow{OA_2}$. where point $A_1$ belongs to the square $F_1$, and point $A_2$ belongs to the square $F_2$. Find the smallest and the largest possible values of the area of the figure $F$ depending on the relative position of the squares $F_1$ and $F_2$.

1991 Ukraine MO grade IX P5

On the sides of an arbitrary triangle $ABC$, the parallelograms $APQC$, $BMNC$, $AEFB$ are constructed outside it. that quadrilateral $AECK$ is also a parallelogram ($K$ is the point of intersection of the lines $PQ$ and $MN$). Prove that the area of $AEFB$ is equal to the sum of the areas of the parallelograms $APQC$ and $BMNC$.

1991 Ukraine MO grade X P4

In space, the lines $a, b$, and $c$ are given. On the line $a$ is taken a point $M_0$. From the point $M_0$ is drawn perpendicular on the line $b$ that intersects it at the point $M_1$. From the point $M_1$ is drawn perpendicular on the line $c$ that intersects it at the point $M_2$. From the point $M_2$ is drawn perpendicular on the line $a$ that intersects it at the point $M_3$, etc. Prove that if $M_6= M_0$. then $M_3 = M_0$.

1991 Ukraine MO grade X P8

On the plane $\alpha$ arbitrarily chosen point $A$ and a circle. From each point $B$ of the circle draw perpendicular on the plane $\alpha$ the length of which is equal to the square length of the segment $AB$ (all perpendiculars are drawn on same one semiplane wrt plane $\alpha$). Prove that the ends of these perpendiculars lie in one plane.

1991 Ukraine MO grade XI P1

On each vertex of a cube, there is one fly. All eight of them buzz off, then return to the vertices of the cube in random order (but still one per vertex). Prove that there are three flies such that the triangles formed from their initial positions and from their final positions are congruent

1991 Ukraine MO grade XI P3

On the plane are given three rays with a common start, which divide the plane into three angles whose sum is $360^o$. Inside each angle is marked a point. Construct with a compass and a ruler a triangle whose vertices lie on these rays (one on each ray) and the sides of which pass through the marked points.

1991 Ukraine MO grade XI P5

On the plane $\alpha$ arbitrarily chosen point $A$ and a circle. From each point $B$ of the circle draw perpendicular on the plane $\alpha$ the length of which is equal to the square length of the segment $AB$ (all perpendiculars are drawn on same one semiplane wrt plane $\alpha$). Prove that the ends of these perpendiculars lie in one plane.

1992 Ukraine MO grade VIII P2

The quadrilateral $ABCD$ is inscribed in a circle, and the center $O$ of this circle lies inside the quadrilateral. It is known that $\angle AOB  +  \angle COD = 180^o$. Prove that the sum of the lengths of the perpendiculars drawn from the point $O$ on the sides of the quadrilateral is equal to half the perimeter of the quadrilateral $ABCD$.

1992 Ukraine MO grade VIII P5

On the plane are given four points $A, B, C$ and $D$. It is known that from points $A,C,D$, the closest to $B$ is the point $A$ and from points $A, B, C$ the closest to $D$ is the point $C$. Prove that the segments $AB$ and $CD$ do not have common points.

1992 Ukraine MO grade IX P4

Point $A$ is selected on a given circle, and point $D$ is selected inside the circle. For each triangle $ABC$, whose vertices $B$ and $C$ lie on this circle, and the side $BC$ passes through the point $D$, we construct the intersection point $M$ of its medians. Find the locus of all such points $M$.

1992 Ukraine MO grade X P2

Let $AA_1$, $BB_1$, $CC_1$ be the angle bisectors of triangle $ABC$. Prove that the equality $\overrightarrow{AA_1}+ \overrightarrow{BB_1} + \overrightarrow{C C_1}= \overrightarrow{0}$ holds if and only if triangle $ABC$ is equilateral.

1992 Ukraine MO grade X P6

Point $A$ and $B$ lie on the sides of a convex polygon F, and the points $A_1$ are such that $\overrightarrow{AB} = \overrightarrow{BA_1}$. Denote by $F_1$ a convex polygon of the smallest area containing a polygon $F$ and point $A_1$. Prove that the area of the polygon $F_1$ is not greater than twice the area of the polygon $F$.

1992 Ukraine MO grade XI P2

Let $ABC$ be an arbitrary triangle. Through the point $K$, taken on side of the $AB$, is drawn straight parallel to $AC$ intersecting side $BC$ at the point $L$, and a line parallel to the $BC$ intersecting side $AC$ at the point $M$. At what position of the point $K$ is the area of the triangle $KML$ will be the largest? What woulf this area be equal to if the area of triangle $ABC$ is equal to $S_0$?

1992 Ukraine MO grade XI P6

Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of triangle $ABC$. Prove that the equality $\overrightarrow{AA_1}+ \overrightarrow{BB_1} + \overrightarrow{C C_1}= \overrightarrow{0}$ holds if and only if triangle $ABC$ is equilateral.

1992 Ukraine MO grade XI P7

The base of the pyramid $SABCD$ is a rectangle $ABCD$. The edge $SA$ is perpendicular to the plane of the base. The plane passing through the vertex $A$ perpendicular to the edge $SC$, intersects the sides of edges $SB$, $SC$, $SD$ at points $B_1$, $C_1$, $D_1$, respectively. Prove that a sphere can be circumscribed around the polyhedron $ABCDB_1C_1 D_1$ .

1993 Ukraine MO grade VIII P3

Use a compass and a ruler to reconstruct triangle $ABC$ given its vertex $A$, the midpoint of the side $BC$ and the foot of the perpendicular drawn from point $B$ on the bisector of the angle $\angle BAC$.

1993 Ukraine MO grade VIII P4

Given an isosceles triangle $ABC$ ($AC = BC$). Circle $S$ with center at point $O$ it tangent to the line $BC$ at point $B$ and to the extension of the side $AC$ beyond point $C$ at point $D$. Prove that theintersection point of lines $AB$ and $DO$ lies on the circle $S$

1993 Ukraine MO grade IX P2

Use a compass and a ruler to reconstruct triangle $ABC$ given its vertex $A$, the midpoint of the side $BC$ and the foot of the perpendicular drawn from point $B$ on the bisector of the angle $\angle BAC$.

1993 Ukraine MO grade IX P7

On the plane is given a circle $\omega_1$ and, a straight line $\ell$ and a point $M$ on it. For each circle $\omega_2$, which tangent to the line $\ell$ at the point $M$, we construct the point intersection $X$ of the common external tangents (if any) to circles $\omega_1$ and $\omega_2$. Prove that the set of points $X$ lies on two lines.

1993 Ukraine MO grade X P4

In the triangle $ABC$, $\angle A=\angle B=72^o$. The angle bisector $AN$ and the median $AM$ intersect the angle bisector $BL$ at points $E$ and $D$ respectively. Prove that $\frac{LE}{DE} =\frac{BL}{LD}$.

1993 Ukraine MO grade X P6

Two lines divide the square into four figures of equal area. Prove that the points of intersection of these lines with the sides of the square are the vertices of the new square.

1993 Ukraine MO grade X P8

Is there a convex pentagon $F$, other than a regular one, such that the pentagon $G$ whose vertices are the interior intersection of the diagonals of $F$, is similar to $F$?

1993 Ukraine MO grade XI P4

Given two perpendicular lines $a$ and $b$. They intersect three given different parallel planes at points $A_1$ and $B_1$, $A_2$ and $B_2$, $A_3$ and $B_3$ respectively. Prove that three spheres constructed by having as diameters the segments $A_1B_1$,$A_2B_2$,$A_3B_3$ intersect at one circle.

1993 Ukraine MO grade XI P7

Given two unequal tangent circles . In the larger circle, an arbitrary diameter $AB$ is drawn and the lengths of the tangents drawn from points $A$ and $B$ to the smaller circle are drawn. Prove that the sum of the squares of the lengths of these tangents does not depend on the choice of the diameter $AB$.

1994 Ukraine MO grade VIII P2

In the trapezoid $ABCD$ is inscribed a circle , tangent to the sides $AB$ and $CD$ at points $E$ and $F,$ respectively. Prove that $AE\cdot EB = CF\cdot FD$.

1994 Ukraine MO grade VIII P5

Point $A$ is given on the side of the acute angle. Construct a point $M$ on this side such that the distance from it to point $A$ is equal to the distance from $M$ to the other side of the angle.

1994 Ukraine MO grade VIII P6

The frame of the cube with edge $1$ was greased with honey. Which is the smallest distance must the beetle crawl to lick all the honey? (The bettles crawls along edges and starts from a vertice)

1994 Ukraine MO grade IX P4

The bisectors of the external angles of a convex quadrilateral form a new quadrilateral. Prove that the sum of the diagonals of the new quadrilateral is not less than the perimeter of the original quadrilateral.

1994 Ukraine MO grade IX P6

Inside the acute-angled triangle $ABC$, point $D$ is taken such that $\angle ADB =180^o - \angle ABC$, and $\angle  ADC =180^o -  \angle  ACB$. Prove that point $D$ lies on the median $AM$ of triangle $ABC$.

1994 Ukraine MO grade IX P8 , X P8

A convex polygon and a point $O$ inside it are given on the plane. Prove that for any natural $n\ge 2$ on the boundary of this polygon there are n different points $A_1,A_2,...,A_n$ such that $$\overrightarrow{OA_1}+\overrightarrow{OA_1}+...+\overrightarrow{OA_n}=\overrightarrow{0}.$$

1994 Ukraine MO grade X P4

A point $D$ was chosen inside the acute-angled triangle $ABC$ so that $\angle DAC = \angle DBC$. Let $K$ and $L$ be the feet of the perpendiculars drawn from the point $D$ on $AC$ and $BC$, respectively. Prove that the midpoints of the segments $AB$, $CD$ and $KL$ lie on the same line.

1994 Ukraine MO grade X P5

A convex quadrilateral, whose area is equal to one, has two parallel sides. Find the smallest possible value of the length of the greater diagonal of this quadrilateral.

1994 Ukraine MO grade XI P3

Two circles of radii $R$ and $r$ ($R> r$) touch internally at the point $M$, and the chord $AB$ of the larger circle touches the smaller circle. What is the greatest value of the perimeter of the triangle $ABM$?

1994 Ukraine MO grade XI P7

The points $P, Q$, and $S$ are given in space. Two rays are drawn from the points $P$ and $Q$, and each of the rays drawn from the point $P$, intersects both rays drawn from the point $Q$. It is known that intersection points $A$, $B$, $C$, $D$ of their rays form a quadrilateral of unit area and that from the pyramid $SABCD$ some plane can be cut off a quadrangular pyramid $SKLMN$, the base of which $KLMN$ is a rectangle. Prove that the volume of the pyramid $SABCD$ does not exceed $\frac16 PQ$

1995 Ukraine MO grade VIII P5

On the sides $AB$ and $CD$ of the convex quadrilateral $ABCD$, it is possible to take the points $M$ and $P$, respectively. that $MC \parallel AP$ and $MD  \parallel BP$. Prove that the quadrilateral $ABCD$ is trapezoid..

1995 Ukraine MO grade IX P3

In triangle $ABC$, the angle at vertex $B$ is $120^o$. It is known that $AB> BC$ and $M$ the midpoint of the side $AC$. Denote by $P$ the midpoint of the broken line $ABC$ and by $Q$ the intersection point of the lines $BC$ and $PM$. Find the measure of the angle $PQB$.

1995 Ukraine MO grade IX P6

Two circles with the center at the point $O$ are drawn on the plane. Find the locus of the points that are the midpoints of the segments, whose one end lies on the first circle, and the other end lies on the second.

1995 Ukraine MO grade X P2

Two different circles are inscribed in the angle $PCQ$ . Let's mark $A$ the touchpoint of the first circle to the side of the $CP$ , and $B$ the touchpoint of the second circle to the side $CQ$. The circle circumscribed around the triangle $ABC$, intersects agian the first circle at the point $L$, and the second again at the point $K$.The line $CL$ intersects the first circle at the point $M$, and the line $CK$ intersects the second circle at the point $T$. Prove that $AM \parallel BT$.

1995 Ukraine MO grade XI P2

What is the largest number of congurent flat angles that a tetrahedron can have if it is not regular?

1995 Ukraine MO grade XI P8

The sphere inscribed in the tetrahedron $ABCD$ is tangent to the faces $BCD$, $ACD$, $ABD$, $ABC$ at points $A_1$, $B_1$, $C_1$, and $D_1$, respectively. We know that lines $AB_1$ and $BA_1$ intersect. Prove that the lines $CD_1$ and $DC_1$ also intersect.

1996 Ukraine MO grade VIII P1

Find all triangles having integer sidelengths and at least two sidelengths such that each square is equal to the sum of the other two sidelengths.

1996 Ukraine MO grade IX P3

In a convex quadrilateral $ABCD$, the diagonal $AC$ divides in half a segment connecting the midpoints of the sides $AD$ and $BC$. Prove that it divides in half and the diagonal $BD$.

1996 Ukraine MO grade IX P6

The circles $c_1$ and $c_2$ are externally tangent at the point $M$. Lines $\ell_1$ and $\ell_2$ ar tangents to the circles $c_1$ and $c_2$ at the points $A_1$ and $A_2$, respectively, and intersect at the point $P$. The point $M$ lies in the angle bisector the angle $A_1PA_2$. Prove that the center of the circle circumscribed around triangle $A_1M_A2$, lies on the circle circumscribed around the triangle $A_1PA_2$.

1997 Ukraine MO grade VIII P3

Consider acute-angled triangles $ABC$ and $APQ$, where $P$ and $Q$ lie on the side $BC.$ Prove that the circumcenter of $\triangle ABC$ is closer to line $BC$ than the circumcenter of $\triangle APQ.$

1997 Ukraine MO grade VIII P5

Construct the bisector of a given angle using a ruler and a compass, but without marking any auxiliary points inside the angle.

1997 Ukraine MO grade IX P3

The incircle of a triangle $ABC$ is tangent to its sides $AB,BC,CA$ at $M,N,K,$ respectively. A line $l$ through the midpoint $D$ of $AC$ is parallel to $MN$ and intersects the lines $BC$ and $BD$ at $T$ and $S$, respectively. Prove that $TC=KD=AS.$

1997 Ukraine MO grade IX P7

Triangles $ABC$ and $A_1 B_1 C_1$ are non-congruent, but $AC=A_1 C_1=b,$ $BC=B_1 C_1=a$, and $BH=B_1 H_1$, where $BH$ and $B_1 H_1$ are the altitudes. Prove the inequality:

$a \cdot AB+b \cdot A_1 B_1 \le \sqrt{2}(a^2+b^2).$

1997 Ukraine MO grade X P4

Two regular pentagons $ABCDE$ and $AEKPL$ are placed in space so that $\angle DAK=60^{\circ}$. Prove that the planes $ACK$ and $BAL$ are perpendicular.

1997 Ukraine MO grade X P7

In a parallelogram $ABCD$, $M$ is the midpoint of $BC$ and $N$ an arbitrary point on the side $AD$. Let $P$ be the intersection of $MN$ and $AC$, and $Q$ the intersection of $AM$ and $BN$. Prove that the triangles $BDQ$ and $DMP$ have equal areas.

1997 Ukraine MO grade XI P1

Let $ABCD$ be a parallelogram with $AB=1$. Suppose that $K$ is a point on the side $AD$ such that $KD=1, \angle ABK=90^{\circ}$ and $\angle DBK=30^{\circ}$. Determine $AD$.

1997 Ukraine MO grade XI P8

On the edges $AB,BC,CD,DA$ of a parallelepiped $ABCDA_1 B_1 C_1 D_1$ points $K,L,M,N$ are selected, respectively. Prove that the circumcenters of the tetrahedra $A_1 AKN, B_1 BKL, C_1 CLM, D_1 DMN$ are vertices of a parallelogram.

1998 Ukraine MO grade VIII P5

A line $l$ and points $A,B$ on the same side of $l$ are given in the plane. Construct (with a ruler and a compass) a point $C$ such that the line $l$ intersects $AC$ at $M$ and $BC$ at $N$, where $BM$ is the altitude and $AN$ the median.

1998 Ukraine MO grade IX P1

Is there a triangle in the coordinate plane whose vertices, centroid, orthocenter, incenter and circumcenter all have integral coordinates?

1998 Ukraine MO grade IX P2

A quadrilateral $ABCD$ is inscribed in a circle with diameter $AD$. Using a ruler and a compass, construct a triangle inscribed in the same circle and having the same area as $ABCD$.

1998 Ukraine MO grade IX P6

Prove that the sum of squared lengths of the medians of a triangle does not exceed the square of its semiperimeter.

1998 Ukraine MO grade X P2

Point $M$ is arbitrarily taken on side $AC$ of a triangle $ABC$. Let $O$ be the intersection of the perpendiculars from the midpoints of segments $AM$ and $MC$ to lines $BC$ and $AB$, respectively. For which position of $M$ is the distance $OM$ minimal?

1998 Ukraine MO grade X P6

Let $AB$ and $CD$ be diameters of a circle with center $O$. For a point $M$ on a shorter arc $CB$, lines $MA$ and $MD$ meet the chord $BC$ at points $P$ and $Q$ respectively. Prove that the sum of the areas of the triangles $CPM$ and $MQB$ equals the area of triangle $DPQ$.

1998 Ukraine MO grade XI P2

The altitude $CD$ of triangle $ABC$ meets the bisector $BK$ of this triangle at $M$ and the altitude $KL$ of $\triangle BKC$ at $N$. The circumcircle of triangle $BKN$ meets the side $AB$ at point $P\ne B$. Prove that the triangle $KPM$ is isosceles.

1998 Ukraine MO grade XI P7

Two spheres are externally tangent at point $P$. The segments $AB$ and $CD$ touch the spheres with $A$ and $C$ lying on the first sphere and $B$ and $D$ on the second. Let $M$ and $N$ be the projections of the midpoints of segments $AC$ and $BD$ on the line connecting the centers of the spheres. Prove that $PM=PN$.

1999 Ukraine MO grade VIII P5

Let $N$ be the point inside a rhombus $ABCD$ such that the triangle $BNC$ is equilateral. The bisector of $\angle ABN$ meets the diagonal $AC$ at $K$. Show that $BK=KN+ND$.

1999 Ukraine MO grade IX P4

The bisectors of angles $A,B,C$ of a triangle $ABC$ intersect the circumcircle of the triangle at $A_1,B_1,C_1$, respectively. Let $P$ be the intersection of the lines $B_1C_1$ and $AB$, and $Q$ be the intersection of the lines $B_1A_1$ and $BC$. Show how to construct the triangle $ABC$ by a ruler and a compass, given its circumcircle, points $P$ and $Q$, and the halfplane determined by $PQ$ in which point $B$ lies.

1999 Ukraine MO grade IX P7

Let $M$ be a fixed point inside a given circle. Two perpendicular chords $AC$ and $BD$ are drawn through $M$, and $K$ and $L$ are the midpoints of $AB$ and $CD$, respectively. Prove that the quantity $AB^2+CD^2-2KL^2$ is independent of the chords $AC$ and $BD$.

1999 Ukraine MO grade X P2

Let $M$ be a point inside a triangle $ABC$. The line through $M$ parallel to $AC$ meets $AB$ at $N$ and $BC$ at $K$. The lines through $M$ parallel to $AB$ and $BC$ meet $AC$ at $D$ and $L$, respectively. Another line through $M$ intersects the sides $AB$ and $BC$ at $P$ and $R$ respectively such that $PM=MR$. Given that the area of $\triangle ABC$ is $S$ and that $\frac{CK}{CB}=a$, compute the area of $\triangle PQR$.

1999 Ukraine MO grade X P8

Let $AA_1,BB_1,CC_1$ be the altitudes of an acute-angled triangle $ABC$, and let $O$ be an arbitrary interior point. Let $M,N,P,Q,R,S$ be the feet of the perpendiculars from $O$ to the lines $AA_1,BC,BB_1,CA,CC_1,AB$, respectively. Prove that the lines $MN,PQ,RS$ are concurrent.

1999 Ukraine MO grade XI P3

All faces of a parallelepiped $ABCDA_1B_1C_1D_1$ are rhombi, and their angles at $A$ are all equal to $\alpha$. Points $M,N,P,Q$ are selected on the edges $A_1B_1,DC,BC,A_1D_1$, respectively, such that $A_1M=BP$ and $DN=A_1Q$. Find the angle between the intersection lines of the plane $A_1BD$ with the planes $AMN$ and $APQ$.

1999 Ukraine MO grade XI P8

Let $AA_1,BB_1,CC_1$ be the altitudes of an acute-angled triangle $ABC$, and let $O$ be an arbitrary interior point. Let $M,N,P,Q,R,S$ be the feet of the perpendiculars from $O$ to the lines $AA_1,BC,BB_1,CA,CC_1,AB$, respectively. Prove that the lines $MN,PQ,RS$ are concurrent.

2000 Ukraine MO grade VIII P5

In the triangle $ABC$, the median $BM$ and the angle bisector $BL$ were drawn (points $M$ and $L$ did not coincide). On the line $BM$ mark the point $E$ such that $LE \parallel BC$. From point $E$, the perpendicular $ED$ is drawn on the line $BL$. Prove that $MD \parallel AB$.

2000 Ukraine MO grade IX P2

The circles $\omega_A$ and $\omega_B$ are tangent to the circle $\omega$ at points $A$ and $B$, respectively, and intersect at points $C$ and $D$ ($AB$ is not the diameter of $\omega$). The lines $AB$ and $CD$ are perpendicular. Prove that the radii of the circles $\omega_A$ and $\omega_B$ are the same.

2000 Ukraine MO grade IX P7

The triangle formed by the intersection points of the extensions of the medians of the triangle $ABC$ with the circle circumscribed around the triangle ABC is equilateral. Prove that the triangle $ABC$ itself is equilateral.

2000 Ukraine MO grade X P4

The acute-angled triangle $PNK$ is inscribed in a circle. The diameter $NM$ of this circle intersects the side $PK$ at point $A$. On the smaller of the arcs $PN$, an arbitrary point $H$ is chosen and a circle is circumscribed around the triangle $PAH$, which intersects the lines $MN$ and $PN$ at points $B$ and $D$, respectively. A circle is constructed having the segment $BN$ as the diameter, which intersects the lines $PN$ and $NK$ at the points $F$ and $Q$, respectively. The segment $FQ$ intersects the diameter $MN$ at point $C$. The straight line $CD$ intersects the circle circumscribed around the triangle $PAH$ at point $E$ ($E \ne D$). Prove that the points $H, E, N$ lie on the same line.

2000 Ukraine MO grade X P7

The parallelogram $ABCD$ and the rhombus $AB_1C_1D_1$ have a common angle $A$. It is known that $BD \parallel CC_1$. Let $P$ be the intersection point of of the lines $AC$ and $B_1C_1$ . Let $Q$ be the intersection point of the lines $AC_1$ and $CD$. Prove that the angle $AQP$ is right.

2000 Ukraine MO grade XI P3

Let $AA_1, BB_1, CC_1$ be the altitudes of the acute triangle $ABC$. Denote by $A_2, B_2, C_2$ the points of tangency of the circle inscribed in triangle $A_1B_1C_1$, with sides $B_1C_1$, $C_1A_1$, $A_1B_1$ respectively. Prove that the lines $AA_2$, $BB_2$, $CC_2$ intersect at one point.

2000 Ukraine MO grade XI P6

The tetrahedron $ABCD$ is known to have $AB = AC = AD = BC$, and the sums of the plane angles at vertices $B$ and $C$ are $150^o$. Find the sums of plane angles at vertices $A$ and $D$.

2001 Ukraine MO grade VIII P2

A convex quadrilateral is known to have the values of its angles equal to an integer number of degrees, and the value of one of them is equal to the product of the values of the other three. Prove that this quadrilateral is a parallelogram or an isosceles trapezoid.

2001 Ukraine MO grade VIII P5

On the sides $AB$ and $BC$ of the isosceles triangle $ABC$, in which $\angle B = 20^o$, the points $D$ and $E$ were chosen, respectively, so that $AD = BE = AC$. Find the measure of angle $BDE$ .

2001 Ukraine MO grade IX P3

On the sides $AB$ and $BC$ of the equilateral triangle $ABC$, the points $M$ and $N$ are taken, respectively, so that $MA = NB$. Prove that there exists a point other than point $B$ through which all the circles circumscribed around the triangle BMN thus obtained pass.

2001 Ukraine MO grade IX P8

In the triangle $ABC$,$I$ is the point of intersection of the angle bisectors $AA_1$ and $CC_1$, $M$ is an arbitrary point on the side $AC$. Lines that are parallel to these bisectors and pass through the point $M$ intersect the segments $AA _1, CC_1, AB$ and $CB$ at the points $H, N, P$ and $Q$, respectively. Let $BC = a$, $AC = b$, $AB = c$, and $d_1,d_2, d_3$ be the distances from the points $H,I, N$ to the line $PQ$, respectively. Prove that

$$\frac{d_1}{d_2}+\frac{d_2}{d_3}+\frac{d_3}{d_1}\ge \frac{2ab}{a^2+bc}+\frac{2ca}{c^2+ab}+\frac{2bc}{b^2+ca}$$

2001 Ukraine MO grade X P4

In the triangle $ABC$ is inscribed a circle that touches the sides $AB, BC$ and $CA$ at points $M, N$ and $K$, respectively. The midline of the triangle $ABC$, parallel to $AB$, intersects the line $MN$ at the point $Q$. Prove that $QM = QK$.

2001 Ukraine MO grade X P7

Given an acute triangle $ABC$. Let $AL$ be its bisector, $M$ and $N$ be the midpoints of the sides $AC$ and $AB$, respectively. Prove that $ML + NL> AL$.

2001 Ukraine MO grade XI P4

The acute-angled triangle $ABC$ ($AC \ne BC$) is inscribed in circle $\omega$. Point $N$ is the midpoint of the arc $AC$ that does not contain point $B$, point $M$ is the midpoint of that arc $BC$ that does not contain point $A$, and point $D$ is the point of arc $MN$ such that $DC\parallel NM$. On the arc $AB$, which does not contain the point $C$, arbitrarily marked the point $K$. Let $O,O_1$ and $O_2$ be the centers of the circles inscribed in the triangles $ABC$, $CAK$ and $CBK$, respectively, $L$ is an intersection point of the line $DO$ with the circle $\omega$ ($L\ne D$). Prove that the points $K, O_1, O_2$ and $L$ lie on the same circle

2001 Ukraine MO grade XI P6

The tetrahedron $ABCD$ is known to have $\angle BAC + \angle BAD = \angle ABC +\angle ABD=90^o$. Let $O$ be the center of the circumcircle of triangle $ABC$, $M$ be the midpoint of the edge $CD$. Prove that the lines AB and $MO$ are perpendicular.

2002 Ukraine MO grade VIII P2

Let $ABCD$ be an isosceles trapezoid in which $BC$ is the smaller base, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AD$, respectively, and the segment $BP$ is its altitude. Denote by $Q$ the intersection point of the segments $DM$ and $BN$. Prove that the points $P, Q$ and $C$ lie on the same line.

2002 Ukraine MO grade VIII P5

Given a convex hexagon in which all angles are equal. Prove that the abdolute values of differences in the lengths of its opposite sides are equal to each other.

2002 Ukraine MO grade VIII P6

On the sides of the triangle $ABC$ (angle $B$ is obtuse), equilateral triangles $ABC_1$, $AB_1 C$, and $A_1BC$ are constructed outside it. Let $B_1$ and $C_2$ be such points that $ABC_1B_1$ and $ACB_1C_2$ are rhombuses. Prove that the line $AA_1$ divides the segment $B_2C_2$ in half.

2002 Ukraine MO grade IX P3

Let the point $K$ lie on the side $AB$ of the triangle $ABC$, and the segment $CK$ intersects it's angle bisector $BF$ at such a point $Q$ that $\angle BQC = 2\angle BFA$ and $\angle BAF = 2\angle CQF$. Prove that $KF = FC$.

2002 Ukraine MO grade IX P5

In the acute-angled triangle $ABC$, $\angle ABC = 60^o$, $A_1,B_1,C_1$ are the feet of the altitudes drawn from the vertices $A, B$, and $C$, respectively. On the rays $B_1A_1$ and $B_1C_1$ marked the points $N$ and $M$, respectively, so that they lie outside the triangle $ABC$ and $NA_1 = A_1C_1 = C_1M$. Prove that the points $N, B, M$ lie on the same line.

2002 Ukraine MO grade X P3

An acute-angled triangle $ABC$ is inscribed in a circle. Using a compass and a ruler, construct at least one hexagon inscribed in this circle with an area twice the area of the triangle $ABC$.

2002 Ukraine MO grade X P5

In the triangle $ABC$, $\angle A=2 \angle B$, $M$ is the midpoint of the side $AB$. Prove that $$\frac{4 \cdot CM^2}{AC^2}=5-4\cos^2 \angle A.$$

2002 Ukraine MO grade X P7

The triangle $ABC$ is inscribed in a circle. Points $A_1,B_1,C_1$ are the midpoints of its arcs $BC,CA,AB$, respectively (arcs that do not contain the third vertices of this triangle are considered), and points $A_2,B_2,C_2$ are the touchpoints of the circle inscribed in triangle $ABC$ with the sides $BC,CA,AB$, respectively . Prove that the lines $A_1A_2, B_1B_2$ and $C_1C_2$ intersect at one point.

2002 Ukraine MO grade XI P4

The base of the quadrangular pyramid $SABCD$ is a rhombus $ABCD$. It is known that $\angle SBA +\angle SBC = 180^o$. On the edge $SC$ marked the point $M$ so that $SM = 2MC$. Prove that the plane passing through the line $DM$ and parallel to the line $AC$ intersects the height of the pyramid in its midpoint.

2002 Ukraine MO grade XI P7

On the sides $AB, BC, AC$ of the acute triangle ABC, the points $C_1,A_1,B_1$, were chosen so that $A_1B = A_1C_1$ and $A_1C = A_1B_1$. Let $I_1$ be the center of a circle inscribed in the triangle $A_1B_1C_1$, and let $H$ be the point of intersection of the altitudes of the triangle $ABC$. Prove that the points $B_1, C_1, I_1$ and $H$ lie on the same circle.

2003 Ukraine MO grade VIII P2

A convex pentagon $ABCDE$ has a circle that touches all its sides. It is known that $\angle BAE = \angle DCB = \angle  AED = 90^o$. Find the measure of the angle $ACE$ .

2003 Ukraine MO grade VIII P6

Given a right trapezoid $ABCD$ with the bases $BC$ and $AD$, in which $BC <AD$ and $\angle A = \angle B = 90^o$. It is known that inside this trapezoid there are two points $M$ and $N$ such that the triangles $AMD$ and $BNC$ are equilateral, $\angle CND = 90^o$, and the point $N$ lies inside the triangle $AMD$. Let $P$ be the intersection point of the lines $CN$ and $DM$, and $Q$ be the intersection point of the lines $AB$ and $DN$. Prove that the lines $PQ$ and $CD$ are perpendicular.

2003 Ukraine MO grade IX P3

On the coordinate plane $xOy$ is given such a triangle $ABC$, both coordinates of each of the vertices of which are integers, and inside (not on the boundary) there is no point, both coordinates of which are integers. Prove that the triangle $ABC$ cannot be acute.

2003 Ukraine MO grade IX P8

On the sides of the triangle $ABC$, the rectangles $ABB_1A_1$, $BCC_1B_2$ and $ACC_2A_2$ are constructed inwards. Prove that the lines passing through the vertices $A, B, C$ perpendicular on the lines $A_1A_2$, $B_1B_2$, $C_1C_2$, respectively, intersect at one point.

2003 Ukraine MO grade X P2

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$ so that the diameter $BK$ of the circle $\omega_1$ intersects the circle $\omega_2$ at point $D$ other than $B$, and the diameter $BP$ of the circle $\omega_2$ intersects the circle $\omega_1$ at point $C$ other than $B$, and points $C$ and $D$ lie on opposite sides of the line $AB$. The line $AD$ intersects the circle $\omega_1$ at the point $M$ other than $A$, and the line $AC$ intersects the circle $\omega_2$ at the point $N$ other than $A$. Prove that $MD = DC = CN$.

2003 Ukraine MO grade X P7

Given an acute-angled triangle $ABC$, in which $\angle C=60^o$, and let point $I$ be the center of the circle inscribed in it. On the side $AB$, such points $M$ and $N$ are chosen that $AM = MN = NB$, and on the sides $AC$ and $BC$, such points $P$ and $Q$ are chosen, respectively, that the quadrilateral $CPIQ$ is a parallelogram. Prove that if $\angle MPI =  \angle NQI$, then the triangle $ABC$ is equilateral.

2003 Ukraine MO grade XI P2

Let $r$ be the radius of the inscribed circle of triangle $ABC$, whose ares is equal to $S$. Prove that $$\frac{S}{r^2}\ge 3\sqrt3.$$

2003 Ukraine MO grade XI P3

Let the point $O$ be the center of the sphere circumscribed around the triangular pyramid $SABC$. It is known that $SA + SB=CA + CB$, $SB + SC = AB + AC$, $SC + SA = BC + BA$. Let the points $A_1, B_1, C_1$ be the midpoints of the edges $BC$, $CA$, $AB$, respectively. Calculate the radius of the sphere circumscribed around the triangular pyramid $OA_1B_1C_1$ if $BC = a$, $CA= b$, $AB = c$.

2003 Ukraine MO grade XI P7

The inscribed circle $\omega$ of triangle $ABC$ touches its sides $AB, BC$ and $AC$ at points $K, M$ and $N$, respectively. Let $P$ be the point of intersection of the lines containing the midlines of the triangles $AKN$ and $CMN$, which are parallel to the sides $KN$ and $MN$, respectively. Prove that the circumcircle of the triangle $APC$ is tangent to the circle $\omega$.

2004 Ukraine MO grade VIII P3

On the side $BC$ of the square $ABCD$, a point $M$ different from the vertices is chosen, and a line is drawn through it, which intersects the diagonal $AC$ and the line $AB$ at the points $N$ and $P$, respectively. It is known that $MN = DN$. Find the measure of the angle $MPD$ .

2004 Ukraine MO grade VIII P6

In the triangle $ABC$, $AB> AC$, the point $M$ is the midpoint of the side $BC$, $AL$ is the bisector of the angle $A$. The line passing through the point $M$ perpendicular to the line $AL$ intersects the side $AB$ at the point $D$. Prove that $AD + MC$ is equal to half the perimeter of the triangle ABC .

2004 Ukraine MO grade IX P5

Let $ABC$ be an isosceles triangle with the base $AC$, the point $K$ lies on the side $AB$. After rotating with center at point $K$, point $A$ moves at point $A_1$, which lies on the extension of side $CB$ beyond point $B$, and point $A_1$ moves at point $A_2$ in the midpoint of side $AC$. Using a compass and a ruler, restore the triangle $ABC$, if only the points $A_1$ and $B$ are given.

2004 Ukraine MO grade IX P6

The angle bisector of the acute angle of a right triangle divides the opposite leg at ratio $a: b$ (counting from the vertex of the acute angle to the vertex of the line), $a> b$. Find the ratio of the length of the angle bisector to the length of this leg.

2004 Ukraine MO grade X P3

Given a convex quadrilateral $ABCD$, in which $O$ is the point of intersection of the diagonals. On the segment $AO$ the point $M$ is chosen, and on the segment $DO$ the point N is chosen, such that $BM \parallel CD$ and $CN \parallel AB$. Prove that $MN \parallel  AD$.

2004 Ukraine MO grade X P9

A circle with center $O$ touches the sides of an angle with vertex $A$ at points $B$ and $C$. On the larger arc of this circle with ends $B$ and $C$, a point $M$ (different from points $B$ and $C$) is chosen, which does not belong to the line $AO$. The lines $BM$ and $CM$ intersect the line $AO$ at points $P$ and $Q$, respectively. Let $K$ be the foot of the perpendicular drawn from the point $P$ on the line $AC$, $L$ be the base of the perpendicular drawn from the point $Q$ on the line $AB$. Prove that the lines $OM$ and $KL$ are perpendicular.

2004 Ukraine MO grade XI P2

The perpendicular bisector of the side $AC$ of the acute triangle $BC$ intersects the side $AB$ at the point $P$, and the extension of the side $BC$ beyond point $B$ at the point $Q$. Prove that $\angle PQB = \angle PBO$, where $O$ is the center of the circumcircle of the triangle $ABC$.

2004 Ukraine MO grade XI P3

Given a triangular pyramid $SABC$, the side edge $SA$ of which is perpendicular to the base $ABC$. Two different spheres $\sigma_1$ and $\sigma_2$ pass through the points $A, B, C$ so that each of them touches from the inside to the sphere $\sigma$, the center of which is at the point $S$. Find the radius $R$ of the sphere $\sigma$ if the radii of the spheres $\sigma_1$ and $\sigma_2$ are $r_1$ and $r_2$, respectively.

2005 Ukraine MO grade VIII P2

Let $AD$ be the median of triangle $ABC$, with $\angle ADB = 45^o$ and $\angle ACB = 30^o$. Find the measure of the angle $BAD$.

2005 Ukraine MO grade VIII P8

In the convex quadrilateral $ABCD$, $BC = CD$ and $\angle CBA + \angle DAB> 180^o$. Points $W$ and $Q$, other than the vertices of the quadrilateral, lie on the sides $BC$ and $DC$, respectively, and $AD = QD$ and the lines $WQ$ and $AD$ are parallel. It is known that the point $M$ of intersection of the segments $AQ$ and $BD$ is equidistant from the lines $AD$ and $BC$. Prove that $\angle BWD = \angle ADW$.

2005 Ukraine MO grade IX P3

There are n\ge 3 points on the plane, which do not lie all on one line. For each point $M$ of the plane through denote $f(M)$ the sum of the distances from these $n$ points to $M$. It is known that there is such a point $M_1$ that for each point $M$ the inequality $f (M_1) \le f(M)$ holds. Let $M_2$ be a point such that $f (M_1) =f(M_2)$. Prove that the points $M_1$ and $M_2$ coincide.

2005 Ukraine MO grade IX P8

Chords $AB$ and $CD$, which do not intersect, are drawn in the circle. On the chord$AB$, a point $E$ different from its ends is chosen. Consider an arc with ends $A$ and $B$, which does not contain points $C$ and $D$. Using a compass and a ruler, construct a point $F$ such that $\frac{PE}{EQ}=\frac12$ where $P$ and $Q$ are the points of intersection of the chord $AB$ with segments $FC$ and $FD$, respectively.

2005 Ukraine MO grade X P4

Inside the parallelogram $ABCD$ marked are two different points $P$ and $Q$ such that $\angle ABP =\angle ADP$ and $\angle CBQ = \angle CDQ$, and these points do not lie on the diagonal $AC$. Prove that $\angle PAQ = \angle PCQ$.

2005 Ukraine MO grade X P7

On the perpendicular bisector of the side $AC$ of the acute triangle $ABC$ noted such a point $M$ that $\angle BAC=\angle MCB$ and $\angle ABC + \angle MBC = 180^o$, and the point M lies on one side with the vertex $B$ wrt the straight line AC. Find the measure of the angle $BAC$.

2005 Ukraine MO grade XI P3

A circle $\omega$ with center at point $O$ was circcumscribed around the acute-angled triangle $ABC$, and then a circle $\omega_1$ was circumscribed around the triangle $AOC$ and the diameter $OQ$ was drawn in it. On the lines $AQ$ and $AC$ marked points $M$ and $N$, respectively, so that the resulting quadrilateral $AMBN$ is a parallelogram. Prove that the intersection point of the lines $MN$ and $BQ$ lies on the circle $\omega_1$.

2006 Ukraine MO grade VIII P8

On the side $BC$ of an equilateral triangle $ABC$ arbitrarily marked a point $M$ different from the vertices. Outside the triangle $ABC$ - on the other side of the point $A$ wrt the line $BC$ - such a point $N$ is chosen that the triangle $BMN$ is equilateral. Let the points $P, Q$ and $R$ be the midpoints of the segments $AB, BN$ and $CM$, respectively. Prove that the triangle $PQR$ is equilateral.

2006 Ukraine MO grade IX P2

Given a trapezoid $ABCD$ with bases $BC$ and $AD$. On the sides $AB$ and $CD$, the points $M$ and $N$ were chosen, respectively, so that the segment $MN$ is parallel to the bases of the trapezoid and passes through the point of intersection of its diagonals. Let the segment $DP$ be the altitude of the triangle $DMC$, and the segment $AQ$ be the altitude of the triangle $ABN$. Prove that $AP = DQ$.

2006 Ukraine MO grade IX P8

Given an acute triangle $ABC$. Let $D$ be a point on the side $BC$ different from the vertices, and let the points $P$ and $Q$ be the centers of the circumcircles of the triangles $ABD$ and $ACD$, respectively. Consider all sorts of triangles $APQ$ obtained for all such points $D$. Prove that the circles circumscribed around all these triangles have a common point other than point $A$.

2006 Ukraine MO grade X P4

Given a trapezoid $MPRQ$ with bases $PM$ and $RQ$ ($RQ <PM$). A point $S$ different from the vertices is marked on the side $RM$. Let $O$ be the point of intersection of the bisectors of the angles $MSQ$ and $MPQ$, and $I$ be the center of the inscribed circle of the triangle $PQR$. It is known that the lines $QR$ and $OI$ are parallel. Prove that $SR = OI$.

2006 Ukraine MO grade X P6

Given a convex pentagon $ABCDE$ inscribed in a circle $\omega$, and the diagonal $AD$ is the diameter of this circle, and the diagonals $BE$ and $AC$ intersect at right angles. Let $P$ be the point of intersection of the segments $CE$ and $AD$. Prove that the area of the triangle $APE$ is equal to the sum of areas of triangles $ABC$ and $CDP$.

2006 Ukraine MO grade XI P4

Let $A$ and $B$ be two different points of intersection of two circles $\omega_1$ and $\omega_2$, $C$ is the point of intersection of the tangent to the circle $\omega_1$, drawn through the point $A$, with the tangent to the circle $\omega_2$, drawn through the point $B$. The straight line $AC$ intersects the circle $\omega_2$ at the point $T$, different from A. On the circle $\omega_1$, a point $X$ different from $A$ and $B$ is arbitrarily chosen so that the line $XA$ intersects the circle $\omega_2$ at the point $Y$, other than $A$. Let the line $YB$ intersect the line $XC$ at the point $Z$. Prove that the lines $TZ$ and $XY$ are parallel.

2006 Ukraine MO grade XI P7

A cube $ABCDA_1B_1C_1D_1$ is given in space. Inside this cube is marked an arbitrary point $M$. On the rays $MA$, $MB$, $MC$, $MD$ ,$MA_1$, $MB_1$, $MC_1$ and $MD_1$ noted are,different from $M$, points $A'$, $B'$, $C'$, $D'$, $A_1$,$B_1'$,$C_1'$,$D_1'$ and $D_1'$, respectively, so that $A'B'C'D'A_1'B_1'C_1'D_1'$ is a parallelepiped (a prism whose faces are parallelograms). Prove that this parallelepiped $A'B'C'D'A_1'B_1'C_1'D_1'$ is a cube.

2007 Ukraine MO grade VIII P3
In an isosceles triangle $ABC$ ($AB = BC$), $\angle ABC = 40 ^o$. On the sides $AB$and $BC$, the points $M$ and $N$ are selected, respectively. It turned out that the segment $MN$ is perpendicular to the side $BC$ and is equal to half of the side $AC$. Prove that $CM = AC$.

Gogolev Andrew

2007 Ukraine MO grade VIII P8
On each side of the triangle $ABC$ on the outside are constructed equilateral triangles: $AB {{C} _ {1}}$,  $A {{B} _ {1}} C$and ${ {A} _ {1}} BC$. Through the midpoints of the segments ${{A} _ {1}} {{B} _ {1}}$, ${{B} _ {1}} {{C} _ {1}}$ and ${{C} _ {1}} {{A} _ {1}}$ held lines perpendicular to the sides $AB$, $BC$ and $AC$, respectively. Prove that the drawn lines intersect at one point.

Alexey Chubenko

2007 Ukraine MO grade IX P4
Inside the triangle $ABC$ with angles $\angle C = 90^o$ and $\angle A = 60^o$ there is a point $O$such that $\angle AOB = 12^o$, $OC = 1$, $OB = 4$. Find the length of the segment $AO$.

Alexey Chubenko

2007 Ukraine MO grade IX P5
In the convex heptagon $ABCDEFG$ the segments are parallel: $AC$ and $EF$, $BD$ and $FG$, $CE$ and $GA$, $DF$ and $AB$], $EG$ and $BC$ and $FA$ and $CD$. Prove that the segments $GB$and $DE$ are also parallel.

Turkevich Edward

2007 Ukraine MO grade X P2
Triangle $ABC$ is acute-angled.$BB_{1}$ and $CC_{1}$ are it's altitudes.$P \in BB_{1}$, $Q \in CC_{1}$, $\measuredangle{APC}=\measuredangle{AQB}=\frac{\pi}{2}$, inradiuses of $\triangle{APC}$ and $\triangle{AQB}$ are equal.Prove that  $AB=AC$.

Alexei Klurman

2007 Ukraine MO grade X P7
In an acute triangle $ABC$, $\angle ABC=60^o$. Point $D$ belongs to side $AC$. Prove the inequality: $\sqrt {3} BD \le AC + \max \{AD, DC \}$

Alexey Chubenko

2007 Ukraine MO grade XI P8 (also)
Triangle $ABC$ is acute-angled.$M$ is a midpoint of bisector $AA_{1}$.$P$ and $Q$ are points on $MB$ and $MC$ respectively.$\measuredangle{APC}=\measuredangle{AQB}=\frac{\pi}{2}$.Prove that $A_{1}PMQ$ is cyclic.

Alexei Klurman

2008 Ukraine MO grade VIII P3
On the side $BC$ of the triangle $ABC$ mark the point $M$ so that $BM = AC$. The point $H$ is the base of the perpendicular drawn from the point $B$ to the line $AM$. It is known that $BH = CM$ and $\angle MAC = 30^o$. Find the degree measure of the angle $\angle ACB$.

Alexei Klurman

2008 Ukraine MO grade VIII P7
In the quadrilateral $ABCD$, the diagonals $AC$  and $BD$ intersect at the point $O$. It is known that the diagonal $BD$ is perpendicular to the side $AD$, $\angle BAD = \angle BCD = 60^o$,  $\angle ADC = 135^o$ . Find the ratio $DO: OB$.

Kryukova Galina

2008 Ukraine MO grade IX P4
The circle inscribed in the triangle $AB$ touches the sides $BC$, $CA$ and $AB$at the points ${{A} _ {1}}$, ${{ B} _ {1}}$ and ${{C} _ {1}}$ respectively. A line perpendicular to $AB$ is drawn through the midpoint of the segment ${{A} _ {1}} {{B} _ {1}}$, through the midpoint of ${{B} _ {1}} { {C} _ {1}}$ is a line perpendicular to $BC$, and through the midpoint of ${{C} _ {1}} {{A} _ {1}}$ is a line perpendicular to to $CA$. Prove that these lines intersect at one point.

Primak Andrew

2008 Ukraine MO grade IX P7
The height $B {{B} _ {1}}$ is drawn in the acute-angled isosceles triangle $ABC$. On the side $BC$, the point $D$ is selected such that $\angle BAD = \angle CB {{B} _ {1}}$. The segments $AD$ and $B {{B} _ {1}}$ intersect at the point $F$. Through the point $B$, perpendicular to the side $AB$, a line $l$ was drawn, which intersects with the line $CF$ at the point $K$. Prove that the line $DK$ intersects the segment $BF$ in its midpoint.

Primak Andrew

2008 Ukraine MO grade X P2
On the extension of the side $BC$ of the parallelogram $ABCD$ to the point $C$ such a point $K$ is chosen that the triangle $CDK$ is isosceles with the base, and on the extension of the side $DC$ for the point $[C$ is chosen such a point $L$ that the triangle $CBL$ is isosceles with the base $CL$. The bisectors of the angles $\angle LBC$ and $\angle CDK$ intersect at the point $Q$. Find the radius of the circle circumscribed  around the triangle $ALK$ if $\angle BQD = \alpha$ and $KL = a$.

Zhidkov Sergey

2008 Ukraine MO grade X P8
Let $H$ be the point of intersection of the heights of an acute triangle $ABC$, the points ${{A} _ {1}}$, ${{B} _ {1}}$, ${ {C} _ {1}}$ - the midpoint of the sides $BC$, $CA$ and $AB$ respectively. Let ${{A} _ {2}}$ and ${{C} _ {2}}$ be such points that ${{A} _ {2}} A \bot AC$ and ${{A} _ {2}} {{C} _ {1}} \bot AB$ , ${{C} _ {2}} C \bot AC$ and ${{C } _ {2}} {{A} _ {1}} \bot BC$. Prove the following statements:
a) the midpoint of the segment $BH$ lies on the line ${{A} _ {2}} {{C} _ {2}}$,
b) let the line $B {{B} _ {1}}$ intersect the circle circumscribed around the triangle ${{A} _ {1}} {{B} _ {1}} {{C} _ { 1}}$, at the points ${{B} _ {1}}$ and ${{B} _ {3}}$, then the point ${{B} _ {3}}$ lies on the line ${{A} _ {2}} {{C} _ {2}}$.

Bilokopitov Eugene

2008 Ukraine MO grade XI P3
Given a triangle $ABC$ inside which there is a point $O$ that $\angle BOC = 90^o$ and $\angle BAO = \angle BCO$. The points $M$ and $N$ are the midpoints of the sides $AC$ and $BC$, respectively. Prove that the angle $\angle OMN$ is right.

Shepelska Barbara

2008 Ukraine MO grade XI P8
In an acute-angled triangle $ABC$ points ${{A} _ {0}}$, ${{B} _ {0}}$, ${{C} _ {0}}$ - bases of heights. Inside the triangle are marked such points ${{A} _ {1}}$, ${{B} _ {1}}$, ${{C} _ {1}}$ that $\angle {{A} _ {1}} BC = \angle {{A} _ {1}} AB$, $\angle {{A} _ {1}} CB = \angle {{A} _ {1}} AC$, $\angle {{B} _ {1}} CA = \angle {{B} _ {1}} BC$, $\angle {{B} _ {1} } AC = \angle {{B} _ {1}} BA$, $\angle {{C} _ {1}} BA = \angle {{C} _ {1}} CB$, $\angle {{C} _ {1}} AB = \angle {{C} _ {1}} CA$. The points ${{A} _ {2}}$, ${{B} _ {2}}$ and ${{C} _ {2}}$ are the midpoints of the segments $A {{ A} _ {1}}$, $B {{B} _ {1}}$ and $C {{C} _ {1}}$, respectively. Prove that the lines ${{A} _ {0}} {{A} _ {2}}$, ${{B} _ {0}} {{B} _ {2}}$ and ${{C} _ {0}} {{C} _ {2}}$ intersect at one point.

Bilokopitov Eugene

2009 Ukraine MO grade VIII P4
In the triangle $ABC$ given that $\angle ABC = 120^\circ .$ The bisector of $\angle B$ meet $AC$ at $M$ and external bisector of $\angle BCA$ meet $AB$ at $P.$ Segments $MP$ and $BC$ intersects at $K$. Prove that $\angle AKM = \angle KPC .$

Zhidkov Sergey

2009 Ukraine MO grade VIII P6
In acute-angled triangle $ABC,$ let $M$ be the midpoint of $BC$ and let $K$ be a point on side $AB.$ We know that $AM$ meet $CK$ at $F$. Prove that if $AK = KF$ then $AB = CF$.

Alexei Klurman

2009 Ukraine MO grade IX P4, grade X P3
In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$

Alexei Klurman

2009 Ukraine MO grade IX P8
In the trapezoid $ABCD$ we know that $CD \perp BC,$ and $CD \perp AD .$ Circle $w$ with diameter $AB$ intersects $AD$ in points $A$ and $P,$ tangent from $P$ to $w$ intersects $CD$ at $M.$ The second tangent from $M$ to $w$ touches $w$ at $Q.$ Prove that midpoint of $CD$ lies on $BQ.$

Zhidkov Sergey

2009 Ukraine MO grade X P8
Let $ABCD$ be a parallelogram with $\angle BAC = 45^\circ,$ and $AC > BD .$ Let $w_1$ and $w_2$ be two circles with diameters $AC$ and $DC,$ respectively. The circle $w_1$ intersects $AB$ at $E$ and the circle $w_2$ intersects $AC$ at $O$ and $C$, and $AD$ at $F.$ Find the ratio of areas of triangles $AOE$ and $COF$ if $AO = a,$ and $FO = b .$

Zhidkov Sergey

2009 Ukraine MO grade XI P3
In triangle $ABC$ let $M$ and $N$ be midpoints of $BC$ and $AC,$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PBC = \angle PCA .$ Prove that if $\angle PNA = \angle AMB,$ then $ABC$ is isosceles triangle.

Alexei Klurman

2010 Ukraine MO grade VIII P3
Point $P$ lies inside the triangle $ABC$. The centers of the circumcircles of triangles $PBC, P AC,P AB$ are $O_A, O_B, O_C$, respectively. Denote by $O_P$ , the center of the circumcircle of the triangle $O_AO_BO_C$. Prove that the point P satisfies the condition $O_P = P$ if and only if $P$ is the orthocenter $\vartriangle ABC$.

Bilokopitov Eugene

2010 Ukraine MO grade VIII P8
Inside an isosceles triangle $ABC$ with base $BC$ and acute angle at the vertex,  mark point $P$ such that $\angle BP C = 2\angle  BAC$. Let $K$ be the foot of the perpendicular dropped from $A$ to the line belonging to the bisector of the angle adjacent to the angle $\angle BPC$. Prove that $BP + PC = 2AK$.

Serdyuk Nazar

2010 Ukraine MO grade IX P3
Given an acute triangle $ABC$. On the perpendicular bisectors of  its sides $AB$ and $BC$ respectively, the points $P$ and $Q$ were noted, and $M$ and $N$ were their projections on the side $AC$ (see fig.). It turned out that $2MN = AC$. Prove that the circumcircle of the triangle $PBQ$ passes through the center of the circumcircle of  the triangle $ABC$.

Nagel Igor

2010 Ukraine MO grade IX P6
Around the acute angle triangle $ABC$ circumscribe a circle. The chord $AD$ is the bisector of the angle of the triangle and intersects the side BC at the point $L$, the chord $DK$ is perpendicular to its side $AC$ and intersects it at the point $M$. Find the ratio $\frac{AM}{MC}$ if   $\frac{BL}{LC}=\frac12$

Tooth Vladimir

2010 Ukraine MO grade X P4
Point $P$ lies inside triangle $ABC$. The centers of inscribed circles in triangles $PBC,PAC, PAB$ are denoted by $I_A, I_B, I_C$, respectively. Denote by the $I_P$  the center of the inscribed circle of the triangle $I_AI_BI_C$. Prove that for a point $P$ that satisfies the condition $I_P = P$, the equalities hold : $AP - BP = AC - BC$, $BP - CP = BA - CA$, $CP - AP = CB - AB$.

Bilokopitov Eugene

2010 Ukraine MO grade X P7
On the sides $AB$ and $BC$ of the triangle $ABC$ we chose the points $K$ and $M$, respectively, so that $AK = KM= MC$. Let $N$ be the point of intersection of the lines $AM$ and $CK, P$ the foot of the perpendicular dropped from the point $N$ on the line $KM$, and $Q$ is a point of the segment $KM$ such that $MQ= KP$. Prove that the inscribed circle of triangle $KMB$ touches the side $KM$ at the point $Q$.

Nagel Igor

2010 Ukraine MO grade XI P3
Inside the parallelogram $ABCD$, points $P$ and $Q$ are marked, which are symmetric with respect to the point of intersection of the diagonals. Prove that the circles circumscribed around the triangles $ABP, CDP, BCQ$ and $ADQ$ have a common point.

Serdyuk Nazar

2010 Ukraine MO grade XI P6
In a convex $ABCD$  the angles $\angle ABC$ and $\angle BCD$ are not less than $120^o$. Prove that $AC + BD> AB + BC + CD$.

Bogdansky Victor

2011 Ukraine MO grade VIII P2
In triangle $ABC$, angle $A$ is twice as large as angle $B, CD$ is the bisector of angle $C$. Prove that $BC = AC + AD$.

Rozhkova Maria

2011 Ukraine MO grade VIII P8
In the parallelogram $ABCD$, $\angle ABC = 105^o$. It is known that inside this parallelogram there is such a point $M$ that the triangle $BMC$ is equilateral and $\angle CMD = 135^o$. Let the point $K$ be the midpoint of the side $AB$. Find the measure of the angle $BKC$.

Vyacheslav Yasinsky

2011 Ukraine MO grade IX P4
On the sides $XY, Y Z .ZX$ of the triangle $XYZ$ mark the points $C,E,A$ respectively . On the segments $AX, CY ,EZ$, respectively, mark the points $B, D , F$ in such a way that $BC \parallel AD, DE \parallel  CF, AF \parallel  BE$. Can the lines $XF, YB$ and $ZD$ intersect at one point?

Vyacheslav Yasinsky

2011 Ukraine MO grade IX P6
In the triangle $ABC$, the point $M$ is the middle of the side $BC$, on the side $AB$ mark the point $N$ so that $NB = 2AN$. It turned out that $\angle CAB = \angle CMN$. Why is the ratio $\frac{AC}{BC}$?

Veklich Bogdan

2011 Ukraine MO grade X P4
Through the point $F$, located outside the circle $k$, drew a tangent $FA$ to this circle  and the secant $FB$, which intersects $k$ at the points $B$ and $C$ (C lies between $F$ and $B$). Through point $C$ draw a tangent to the circle $k$, which intersected the segment $FA$ at the point $E$. The segment $FX$ is the bisector of the triangle $AFC$. It turned out that the points $E, X$ and $B$ lie on the same line. Prove that the product of the lengths of two sides triangle $ABC$ is equal to the square of the length of the third side.

Bezverkhnev Yaroslav

2011 Ukraine MO grade X P7
Let $ABC$ be triangle with $AC>BC>AB$. On the sides $BC$ and $AC$, the points $D$ and $K$ were chosen respectively such that $CD=AB, AK=BC$. Points $F ,L$ are the midpoints of the segments $BD ,KC$  respectively. Points $R,S$ are the midpoints of the sides $AC,AB$  respectively. The segments $SL$ and $FR$ intersect at the point $O$ with $\angle SOF = 35^o$. Find the measure of $\angle BAC$.

Zhidkov Sergey

2011 Ukraine MO grade XI P4
The trapezoid $ABCD$ with bases $AD$ and $BC$ is given. On the side of the $CD$ is arbitrary noted the point $F, E$ is the point of intersection of the lines $AF$ and $BD$. On the side $AB$, mark the point $G$ so that that $EG \parallel AD$. Denote by $H$ the point of intersection of the lines $CG, BD$, by $I$ the point of intersection of the lines $FH ,AB$. Prove that the lines $CI, FG$ and $AD$ intersect at one point.

Vyacheslav Yasinsky

2011 Ukraine MO grade XI P5

A circle that passes through the vertices $A$ and $B$ of the triangle $ABC$, touches the side $BC$ at point $B$ and again crosses the side $AC$ at point $E$. A second circle passes through the vertices $A$ and $C$, touches the side $BC$ at point $C$ and crosses the side $AB$ at point $D$. The segments $BE$ and $CD$ intersect at point $F$. Prove that $\vartriangle BCF$ is isosceles.

Rozhkova Maria

2012 Ukraine MO grade VIII P4

Let the inscribed circle of triangle $ABC$ touch its sides $AB, BC ,CA$ at points $K, N, M$, respectively, and it is known that $\angle MKC =\angle MNA$. Prove that triangle $ABC$ is isosceles.

VA Yasinsky

2012 Ukraine MO grade VIII P7

Let point $I$ be the center of the inscribed circle of triangle $ABC$. On the  side $AB$ is selected such a point $M$ different from the vertices that $BM < BC$, and the circumcribed circle of the triangle $AMI$ intersects the side $AC$ at the point $N$, which does not coincide with points $A$ and $C$. Prove that $BM + CN = BC$.

VA Yasinsky

2012 Ukraine MO grade IX P3

Given a triangle $ABC$. Let $I_A$ be the center of a circle tangent to the side $BC$ and to extensions of sides $AB$ and $AC$ beyond points $B$ and $C$, respectively. Prove that the points $B, C$ and the centers of the circumscribed circles of triangles $ABI_A$ and $ACI_A$ lie on the same circle.

VA Yasinsky

2012 Ukraine MO grade IX P6

Given a triangle $ABC$, in which $\angle C=90^o$, $AC<BC$. On the side $BC$  is marked such a point $K$ that $CK=CA$. Let $D$ be a point of the segment $CK$ such that $\angle DAK=\angle BAK$. The segment $DF$ is the altitude of the triangle $ADB$, and the point $P$ is the foot of the perpendicular drawn from point $A$ on the line $FK$. Prove that $CP=\frac12 (AF+FD+DA)$

IP Nagel

2012 Ukraine MO grade X P3

Let $O$ be the center of the circumcircle of an acute non-isosceles triangle $ABC$. The lines $BO$ and $CO$ intersect the sides $AC$ and $AB$ at points $K$ and $N$, respectively. On the sides $AC$ and $AB$ are taken such different from $K$ and $N$ points $P$ and $T$, respectively, that $OK = OP$   and $ON = OT$. A line parallel to $BK$ is drawn through the point $P$, and a line parallel to $CN$ is drawn through the point $T$, and we denote by $M$ the point of intersection of these lines. Prove that the radii of the circumcircles of the triangles $AMB, BMC$ and $CMA$ are equal.

IP Nagel

2012 Ukraine MO grade X P7

Two circles $\omega_1$ and $\omega_2$, which do not have common points, are inscribed in the angle $\angle BAC$, $B\in \omega_1, C\in \omega_2$, and the radius of the circle $\omega_1$ is less than the radius of the circle $\omega_2$. The line $BC$ intersects the circles $\omega_1$ and $\omega_2$ for the second time at the points $K$ and $N$, respectively. Lines $AK$ and $AN$ pass, respectively, through the points $P\in \omega_1$ and $M\in \omega_2$, other than $K$ and $N$. Prove that point $A$ lie on a line passing through the centers of the circumcircles of the triangles $ACM$ and $ABP$.

IP Nagel

2012 Ukraine MO grade XI P4

Let $SABC$ be such a triangular pyramid that for the point $M$,  its median intersection of the face $ABC$, hold the inequalities $MA>1, MB>1$ and $MC>1$. Prove that $SA+SB+SC>3$ 

VA Yasinsky

2012 Ukraine MO grade XI P6

Let $H$ be the point of intersection of the altitudes of an acute isosceles triangle $ABC$, $M$ is the midpoint of the side $AB$, $N$ is the midpoint of the side $AC$. Denote by, respectively, $P$ and $Q$ the points of intersection of the rays $MH$ and $NH$ with the circumcircle of the triangle $ABC$. Prove that the lines $BQ, AH$ and $CP$ intersect at one point or in parallel.

VA Yasinsky

2013 Ukraine MO grade VIII P4

Let $M$ be the midpoint of the lateral side $AB$ of  trapezoid $ABCD$, $O$ be  intersection point of its diagonals, and $AO = BO$. The point $P$ was marked on the ray  $OM$ such that  that $\angle PAC = 90^o$. Prove that  $\angle AMD =  \angle APC$.

2013 Ukraine MO grade VIII P6

Inside an acute-angled triangle $ABC$ denote a point $Q$ such that $\angle QAC = 60^o$, $\angle QCA = \angle  QBA = 30^o$. Let points $M$ and $N$ are the midpoints of the sides$AC$ and $BC$ respectively. Find the measure of the angle $\angle  QNM$.

2013 Ukraine MO grade IX P3

Let $M$ be the midpoint of the side $BC$ of an acute-angled triangle $ABC$, in which $AB \ne AC$, $O$ is the center of its circumcircle. Draw from point $M$ the perpendicular on $MP$ and $MQ$ on the sides $AB$ and $AC$, respectively. Prove the line that passes through the midpoint of the segment $PQ$ and the point $M$, is parallel to the line $AO$.

2013 Ukraine MO grade IX P8

Around an acute-angled triangle $ABC$, in which $AB < BC < AC$, circumscribes a circle $\omega$ with center $O$. Denote $I$ the center of  inscribed circle of the given triangle, and $M$ the midpoint of the side $BC$ . Let $Q$ be the point symmetric to point $I$ wrt $M$, ray $OM$ intersects the circle $\omega$ at the point $D$, and the ray $QD$ intersects the circle $\omega$ for the second time at the point $T$. Prove that $\angle ACT = \angle DOI$.

2013 Ukraine MO grades X P2, XI P2
Let $M$ be the midpoint of the side $BC$ of $\triangle ABC$. On the side $AB$ and $AC$ the points $E$ and $F$ are chosen. Let $K$ be the point of the intersection of $BF$ and $CE$ and $L$ be chosen in a way that $CL\parallel AB$ and $BL\parallel CE$. Let $N$ be the point of intersection of $AM$ and $CL$. Show that $KN$ is parallel to $FL$.

2013 Ukraine MO grade X P8 
Let $M$ be the midpoint of the internal bisector $AD$ of $\triangle ABC$.Circle $\omega_1$ with diameter $AC$ intersects $BM$ at $E$ and circle $\omega_2$ with diameter $AB$ intersects $CM$ at $F$.Show that $B,E,F,C$ are concyclic.

2013 Ukraine MO grade XI P8    (Iran 2007 TST, Croatia TST 2016 ) 

Let $O$ be the center of the circumcircle of an acute-angled triangle $ABC$. On the segments $OB$ and $OC$, the points $E$ and $F$ were chosen, respectively, so that $BE =OF$. Denote $M$ and $N$ the midpoints of the arcs $AOE$ and $AOF$ of the circumscribed circles of triangles $AOE$ and $AOF$, respectively. Prove that $\angle ENO +\angle FMO = 2\angle BAC$.

2014 Ukraine MO grade VIII P2 , IX P1

Two circles ${{\gamma} _ {1}}$ and ${{\gamma} _ {2}}$ of the same radius intersect at the points $A$ and $B$. The circle $\gamma$, centered at the point $A$, intersects the circle ${{\gamma} _ {1}}$ at the points $C$ and $D$. Prove that the points of intersection of the circles $\gamma$ and ${{\gamma} _ {2}}$ belong to the lines $BC$ and $BD$. 

Yuri Biletsky

2014 Ukraine MO grade VIII P8

On the line from left to right are the points $A, \, \, D$ and $C$ so that $CD = 2AD$. The point $B$ satisfies the conditions $\angle CAB = 45 {} ^ \circ$ and $\angle CDB = 60 {} ^ \circ$. Find the measure of the angle $BCD$.

Gerasimova Tatiana

2014 Ukraine MO grade IX P8

The acute triangle $ABC$ is inscribed in the circle ${{w} _ {1}}$, $AN$ and $CK$ are its altitudes, $H$ is the orthocenter. The circle ${{w} _ {2}}$, which is circumscribed around $\Delta NBK$, intersects the circle ${{w} _ {1}}$ for the second time at the point $P$. The lines $CA$ and $BP$ intersect at the point $S$. The line $SH$ intersects the circle ${{w} _ {2}}$ for the second time at the point $Q$. Prove that the lines $NQ, \, \, PK$ and $CA$ intersect at one point, or are parallel.

Igor Nagel

2014 Ukraine MO grade X P2, XI P2

The circle $\gamma$ is circumscribed around the acute triangle $ABC$, $AD$ and $AL$ - its altitude  and bisector, respectively. Denote by $W$, $T$, ${A}'$ the second point of intersection with the circle $\gamma$ lines $AL$, $WD$, $TL$  respectively. Prove that $A {A}'$ is the diameter of the circle $\gamma$. 

Maria Rozhkova

2014 Ukraine MO grade X P8

The inscribed circle of the triangle $ABC$ touches its sides $AB$, $BC$ and $AC$ at the points $N$, $K$, $P$ respectively. It is known that $AB> BC$ and the bisectors of the angles $A$ and $C$ intersect the line $NK$ at the points $Q$ and $T$, respectively. Denote by $S$ - the point of intersection of the lines $AQ$ and $TP$, and by $F$ - the point of intersection of the lines $CT$ and $PQ$. Prove that the lines $NK$, $SF$ and $AC$ intersect at one point.

Igor Nagel

2014 Ukraine MO grade XI P7

The inscribed circle of an acute triangle $ABC$ touches the sides $BA$ and $AC$ at the points $K$ and $L$, respectively. The altitude $AH$ intersects the bisectors of the angles $B$ and $C$ at the points $P$ and $Q$, respectively. Cirumscribed circles of triangles $KPB$ and $LQC$ are denoted by ${{w} _ {1}}$ and ${{w} _ {2}}$. Prove that if the midpoint of the altitude $AH$ lies outside the circles ${{w} _ {1}}$ and ${{w} _ {2}}$, then the tangents to the circles ${ {w} _ {1}}$ and ${{w} _ {2}}$, drawn from this midpoint, are equal. 

Hilko Danilo

2015 Ukraine MO grade VIII P4

Given a trapezoid $ABCD$ with bases $BC$ and $AD$. On the diagonals $AC$ and $BD$ the points $P$ and $Q$ are marked so that $AC$ is the bisector  of $\angle BPD$, and $BD$ is the bisector of $\angle AQC$. Prove that $\angle BPD = \angle AQC$. 

Serdyuk Nazar

2015 Ukraine MO grade VIII P6

In the trapezoid $ABCD$ with perpendicular diagonals of the point $P, N, Q, M$ - the midpoints of the sides $AB, BC, CD, DA$, respectively. Based on $CD$, there is a point $L$, which is different from the point $Q$, for which the angle $MLN$ is right. Find the value of the angle $LPA$. 

Bogdan Rublev

2015 Ukraine MO grade IX P4

An arbitrary point $D$ is selected on the side $BC$ of the acute triangle $ABC$. Let $O$ be the center of the circumscribed circle $\Delta ABC$, and let $Z$ be the point of this circle that is diametrically opposite to the point $A$. Let $X, \, \, Y$ have the following points on the segments $BO, \, \, CO$, respectively, for which the condition holds: $\angle BXD + \angle ABC = 180 {} ^\circ = \angle CYD + \angle ACB$. Prove that the measure of $\angle XZY$ does not depend on the choice of the point $D$.

Hilko Danilo

2015 Ukraine MO grade IX P6

In the triangle $ABC$ on the sides $BC$ and $AB$, the points ${{A} _ {1}}$ and ${{C} _ {1}}$ are selected, respectively. such that $A {{A} _ {1}} = C {{C} _ {1}}$. The segments $A {{A} _ {1}}$ and $C {{C} _ {1}}$ intersect at the point $F$. It turned out that $\angle CF {{A} _ {1}} = 2 \angle ABC$. Prove that $A {{A} _ {1}} = AC$.

Gogolev Andrew

2015 Ukraine MO grade X P2

Inside the right triangle $ABC$, the point $M$ is selected. Let the points ${{M} _ {1}}$, ${{M} _ {2}}$, ${{M} _ {3}}$ be symmetric to it with respect to the sides $BC$, $AC$, $AB$ of the triangle respectively. Prove that the sum of the vectors $\overrightarrow {M {{M} _ {1}}} + \overrightarrow {M {{M} _ {2}}} + \overrightarrow {M {{M} _ {3}} }$ is equal to the sum of the vectors $\overrightarrow {MA} + \overrightarrow {MB} + \overrightarrow {MC}$. 

Teryoshin Dmitry

2015 Ukraine MO grades  X P7, XI P6

In the acute-angled triangle $ABC$, the altitudes $A {{A} _ {1}}$ and $B {{B} _ {1}}$ intersect at the point $H$. Construct two circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers at the points $H$ and $B$ and radii $H {{B} _ {1}}$ and $B {{B} _ {1}}$, respectively. From the point $C$ to the circles ${{w} _ {1}}$ and ${{w} _ {2}}$ we draw tangents that touch these circles at the points $N$ and $K$ other than ${{B} _ {1}}$, respectively. Prove that the points ${{A} _ {1}}$, $N$ and $K$ lie on the same line. 

Igor Nagel

2015 Ukraine MO grade XI P4
In convex quadrilateral $ABCD$ with angles $ABC$ and $BCD$ equal to $120^{\circ}$, $O$ is intersection of diagonals, $M$ is midpoint of $BC$. $K$ is intersection of segments $MO$ and $AD$. It's known that $\angle BKC=60^{\circ}$. Prove that $\angle BKA=\angle CKD = 60^{\circ}$.

Nazar Serdyuk

2016 Ukraine MO grade VIII P2

In an acute-angled triangle $ABC$ with angle $\angle ACB = 60^o$, bisector $BL$ and altitude $BH$ were drawn. The perpendicular from the point $L$ on the side $BC$ is $LD$. Find the angles of $\vartriangle ABC$, if it turns out that $AB\parallel HD$.

Gogolev Andrew

2016 Ukraine MO grade VIII P8

Given a triangle $ABC$, in which $\angle ABC = \angle ACB = 30 {} ^ \circ$. Point $D$ is selected on the  side $BC$. The point $K$ is such that $D$ is the midpoint of $AK$. It turned out that $\angle BKA> 60 {} ^ \circ$. Prove that $3AD <CB$.

Hilko Danilo

2016 Ukraine MO grade IX P2, X P2

The bisector of the angle $\angle ABC$ of the triangle $ABC$ intersects the circumcircle of the triangle at point $K$. The point $N$ lies on the segment $AB$, such that $NK \perp  AB$. Through the midpoint $P$ of the segment $NB$, a line parallel to the line $BC$ intersects line $BK$ at point $T$. Prove that the line $NT$ bisects the segment $AC$.

Igor Nagel

2016 Ukraine MO grade IX P7

Given a triangle $ABC$, in which $AB> AC$. A tangent to the circumcircle of the triangle $ABC$ is drawn through the point $A$. This tangent intersects the line $BC$ at the point $P$. On the extension of the side $BA$ for the point $A$ mark the point $Q$ so that $AQ = AC$. Let $X$ and $Y$ be the midpoints of the segments $CQ$ and $AP$, respectively, and let $R$ belong to the segment $AP$, and $AR = CP$. Prove that $CR = 2XY$.

Vyacheslav Yasinsky

2016 Ukraine MO grade X P8

The triangle $APQ$ and the rectangle $ABCD$ are located on the plane so that the midpoint of the segment $PQ$ belongs to the diagonal $BD$ of the rectangle, and one of the rays $AB$ or $AD$ is the bisector of the angle $PAQ$. Prove that one of the rays $CB$ or $CD$ is the bisector of the angle $PCQ$.

Vyacheslav Yasinsky

2016 Ukraine MO grade XI P2

The circle inscribed in triangle $ABC$, touches its sides $AB, BC$ and CA at points $N, P, K$, respectively. Segment $BK$ intersects the inscribed circle at the point $L$ for the second time. Define the points $T= AL\cap NK$, $Q =CL\cap KP$. Prove that the lines $BK, NQ$ and $PT$ intersect at one point. 

Igor Nagel

2016 Ukraine MO grade XI P7
Triangle $ABC$ is given. Circle $\omega$ with center $Q$ is tangent to side $BC$ and touches the circumcircle of triangle $ABC$ internally at point $A.$ Let $M$ be the midpoint of $BC$ and $N$ be the midpoint of arc $BAC$ of the circumcircle of triangle $ABC.$ Point $S$ is chosen on the segment $BC$ such that $\angle BAM=\angle SAC$. Prove that points $N,$ $Q$ and $S$ are collinear.

M. Plotnikov

2017 Ukraine MO grade VIII P4
$\triangle ABC$. $\angle C = 90^\circ$. Inside $\triangle ABC$ there is point $K$, such that $\angle AKC = 90^\circ$ and $\angle CKB = 2 \angle CAB$. On segment $KB$ there is point $T$, such that $\angle KTC = \angle CAK$.  $P = AK \cap BC$.  Prove that  $\angle TPA = \angle ABC$.

2017 Ukraine MO grade VIII P

Given an acute triangle $ABC$. Let $D$ be a point symmetric to point $A$ wrt $BC$. The lines $DB$ and $DC$ intersect the circumscribed circle $w$ of the triangle $ABC$ for the second time at points $X$ and $Y$ respectively. Assume that the points $X$ and $Y$ lie inside the segments $DB$ and $DC$ respectively. Prove that the center of the circumscribed circle of triangle $XYD$ lies on the circle $w$.

Danilo Hilko

2017 Ukraine MO grade IX P4 (also)
$\omega$ - circle with diameter $AB$ and center $O$. $CD$ - chord perpendicular $AB$. $E$ - middle point of $OC$. Line $AE$ intersects $\omega$  at $F$ and $BC$ at $M$. $L = BC \cap DF$. $\odot (DLM)$ intersects $\omega$ at $K$.  Prove that $KM=MB$

2017 Ukraine MO grade IX P6

Let $w_1, w_2$ be two circles on a plane that do not intersect. Line $AB$ is their common external tangent, $O$ is point of intersection of common internal tangents, $H$ is the foot of perpendicular drawn from $O$ on $AB$. From the point $H$, the tangents $HC, HD$  were drawn to the circles $w_1, w_2$, respectively, different from $AB$. Prove that $HO$ is the bisector of $\angle CHD$. 

Nazar Serdyuk

2017 Ukraine MO grade X P3

In the acute-angled triangle $ABC,$  $\angle ACB= 45^o$, $M$ is the point of intersection of the medians, $O$ is the center of the circumscribed circle. It is known that $OM=1$ and $OM\parallel BC$. Find the length of the side $BC$.

Maxim Black

2017 Ukraine MO grade XI P4

Bob once lived on the plane  of a triangle, in which the orthocenter belonged to the inscribed circle. One day Bogdan deleted the triangle on that plane. After that on it remained  only the inscribed circle of Bob's Triangle, a line containing one of its sides, and its orthocenter. Curious Maxim wants according to these data to  restore triangle with compass and ruler. Help him do it.  

There is no need to study the possibility of construction.

B. Kivva, M. Chaudhari

Жив якось на площині Трикутник Боб, у якого ортоцентр належав вписаному колу. Одного дня розбишака Богдан похазяйнував на тій площині. Після цього на ній лишились намальованими лише вписане коло Трикутника Боба, пряма, що містить одну із його сторін, та його ортоцентр. Допитливий Максим хоче за цими даними відновити Трикутник циркулем та лінійкою. Допоможіть йому це зробити.

Дослідження можливості побудови робити не потрібно.

Б. Ківва, М. Чаудхарі

2017 Ukraine MO grade XI P6
The quadrilateral $ABCD$ is inscribed in a circle $\omega$ with the center $O$ . Its diagonals
intersect at $H$ .  $O_1$ and $O_2$ are the centers of the $\odot (AHD)$ and $\odot (BHC)$ respectively. Line through $H$ intersects $\omega$ at $M_1$ and $M_2$. It also intersects $\odot (O_1HO)$ and $\odot (O_2HO)$ at $N_1$ and $N_2$ respectively. $N_1$ and $N_2$ are lay inside of $\omega$. Prove that $M_1N_1=M_2N_2$.

2018 Ukraine MO grade VIII P4

In the triangle $ABC$ the orthocenter $H$ is marked and the altitude  $AK$ is drawn. The circle $w$ passes through the points $A$ and $K$ and intersects the sides $AB$ and $AC$ at the points $M$ and $N$, respectively. The line passing through the point $A$ parallel to $BC$, for the second time intersects the circumscribed circles of triangles $AHM$ and $AHN$ at the points $X$ and$Y$, respectively. Prove that $XY = BC$.

Danilo Hilko

2018 Ukraine MO grade VIII P6

On the sides $AB$, $BC$, $AC$ of the triangle $ABC$ with $\angle BAC = 120 {} ^ \circ$, mark the points $M$, $K$, $N$ respectively so that $\Delta MKN$ is right, and $AM = 2,017$, $AN = 2,018$. Baron Munchausen claims that $\Delta MKN$ has the smallest perimeter among all right triangles having exactly one vertex on each side of $\Delta ABC$. Isn't the baron wrong?

Maria Rozhkova

2018 Ukraine MO grade IX P2

In an acute-angled triangle $ABC$, the bisector of the angle $A$ intersects the circle circumscribed around this triangle at the point $W$. From the point $W$ on the line $AB$ the perpendicular $WU$ was drawn, and from the center of the inscribed circle $I$ of the same triangle the perpendicular $IP$ was drawn on the line $WU$. Let $M$ be the midpoint of the segment $BC$. Prove that the line $MP$ passes through the middle of the segment $CI$. 

Mykola Moroz

2018 Ukraine MO grade IX P6

In the triangle $ABC$ we denote the points ${{M} _ {1}}$, ${{M} _ {2}}$, ${{M} _ {3}}$ - the midpoints of the sides $BC$, $AC$, $AB$ respectively. The point $K$ is symmetric to ${{M} _ {2}}$ wrt line $BC$, $AH$ is the altitude of triangle $ABC$. Prove that the line $K {{M} _ {3}}$ bisects the segment $H {{M} _ {1}}$. 

Danilo Hilko

2018 Ukraine MO grade X P2

An isosceles obtuse triangle $ABC$ with vertex at point $B$ is given. The perpendicular to the side $BC$ intersects the lines $AC$ and $AB$ at the points $K$ and $M$, respectively. Prove that the point symmetric to the point $A$ wrt $BK$ lies on the line $CM$. 

Anton Trygub

2018 Ukraine MO grade X P8

In a triangle $ABC$, a line that does not coincide with the sides of the triangle and passes through the point $A$ intersects the altitudes $B {{H} _ {2}}$ and $C{H} _ 3$ at points ${{D} _ {1}}$ and ${{E} _ {1}}$ respectively. The points ${{D} _ {2}}$ and ${{E} _ {2}}$ are symmetric to the points ${{D} _ {1}}$ and ${{E} _ {1}}$ wrt the sides $AB$ and $AC$, respectively. Prove that the circles circumscribed around triangles ${{D} _ {2}} AB$ and ${{E} _ {2}} AC$ are tangent. 

Mikhail Plotnikov

2018 Ukraine MO grade XI P2
In acute-angled triangle $ABC$, $AH$ is an altitude and $AM$ is a median. Points $X$ and $Y$ on lines $AB$ and $AC$ respectively are such that $AX=XC$ and $AY=YB$. Prove that the midpoint of $XY$ is equidistant from $H$ and $M$.

Danylo Khilko

2018 Ukraine MO grade XI P8
Given an acute-angled triangle $ABC$, $AA_1$ and $CC_1$ are its angle bisectors, $I$ is its incenter, $M$ and $N$ are the midpoints of $AI$ and $CI$. Points $K$ and $L$ in the interior of triangles $AC_1I$ and $CA_1I$ respectively are such that $\angle AKI = \angle CLI = \angle AIC$, $\angle AKM = \angle ICA$, $\angle CLN = \angle IAC$. Prove that the circumradii of triangles $KIL$ and $ABC$ are equal.

Anton Trygub

2019 Ukraine MO grade VIII P7
Let $ABC$ be a triangle. Points $C_1, A_1$ and $B_1$ are chosen on $AB, BC$ and $CA$, respectively. $K$ is the projetion of $B_1$ onto $A_1C_1$. Points $M$ and $N$ on rays $B_1A$ and $B_1C$ are such that $\angle B_1A_1C_1 = 2\angle KNB_1$ and $\angle B_1C_1A_1 = 2\angle KMB_1$. Prove that $MN$ does not exceed the perimeter of $\vartriangle A_1B_1C_1$.

Anton Trygub

2019 Ukraine MO grade IX P2
Let $M$ be the midpoint of the hypotenuse $AB$ of a right triangle $ABC$. The perpendicular bisector to the hypotenuse $AB$ intersects the side $BC$ at a point $K$. A perpendicular drawn on line $CM$ from a point $K$ intersects the extenstion of the segment $AC$ beyond point $A$ at a point $P.$ Lines $CM$ and $BP$ intersect at a point $T$. Prove that $AC=TB$

Danylo Khilko

2019 Ukraine MO grade IX P8
An acute triangle  $ABC$ is inscribed in a circle $w$ centered at a point $O$. Extensions of its heights, which are drawn from the vertices $A$ and $C$, intersect for the second time $w$ at points $A_0$ and $C_0$ respectively. The line $A_0C_0$ intersects the sides $AB$ and $BC$ at points $A_1$ and $C_1$ respectively. The points $A_2$ and $C_2$ lie on the side such that $A_2O // BC$ and $C_2O // AB$. Let $H$ be the orthocenter of triangle $\vartriangle ABC, T$ be the point of intersection of $A_1A_2$ and $C_1C_2$. Prove that $HT // AC$.

Anton Trygub

2019 Ukraine MO grade X P3

In an acute-angled triangle $ABC$, an inscribed circle with the center $I$ at a point touches the sides $AB, AC$ at the points $C_1,A_1$, respectively. Let  $M$ be the midpoint of $AC, N$ be the midpoint of the arc $ABC$ of the circumcircle of the triangle $ABC, P$ be the projection of the point $M$ on $A_1C_1$. Prove that the points $I,P$ and $N$ lie on the same line.

Anton Trygub

2019 Ukraine MO grade X P6
Given a parallelogram $ABCD$. A circle passing through the vertices $A$ and $D$ intersects the lines  $AB,BD,AC,CD$ at the points $B_1,B_2,C_1,C_2$ respectively. Lines $B_1B_2$ and $C_1C_2$  intersect at a point $K$. Prove that the point is $K$ equidistant from the lines $AB$ and $CD$.

Anton Trygub

2019 Ukraine MO grade XI P4

On a circle with a diameter $AD$ take  points $B,C$ so that $AB=AC$. The point $P$ on the segment $BC$ is chosen in an arbitrary way, and the points $M,N$ on the segments $AB , AC$ respectively, such that the quadrilateral $PMAN$ is a parallelogram. Let $PL$ be the angle bisector of the triangle $MPN$. Line $PD$ intersects $MN$ at a point $Q$. Prove that the points $B,Q,L$ and lie $C$ on the same circle.

Mykhailo Plotnykov, Danylo Khilko

2020 Ukraine MO grade VIII P4  Online

A triangle $ABC$ is such that $\angle A = 75^{\circ}$, $\angle C = 45^{\circ}$. Points $P$ and $T$ on sides $\overline{AB}$, $\overline{BC}$, respectively, such that quadrilateral $APTC$ is cyclic and $CT=2AP$. Let $O$ be the circumcenter of $\triangle ABC$. Ray $TO$ intersects side $\overline{AC}$ at $K$. Prove that $TO=OK$.

2020 Ukraine MO grade IX P4 Online

Let $H$ be the orthocenter of $\triangle ABC$ and $O$ its circumcenter. Suppose that $H$ is the midpoint of the altitude $\overline{AD}$. Line perpendicular to $\overline{OH}$ trough $H$ intersects sides $\overline{AB}$ and $\overline{AC}$ at $P$ and $Q$, respectively. Prove that the midpoints of segments $\overline{BP}$, $\overline{CQ}$ and point $O$ are collinear.

2020 Ukraine MO grade IX P6 Online

Let $ABCD$  be a quadrilateral circumcribed around a circle centered at a point $O$. On the extensions of $AB, AD, CB$ and $CD$ take equal segments $AA_1, AA_2, CC_1$ and $CC_2$ respectively , length which is greater than the length of any side of the quadrilateral $ABCD$. It turned out that points $A_1 , A_2, C_1$ and $C_2$ lie on the same circle with the center at a point $O$. Prove that lines $A_1A_2 , C_1C_2$ and $BD$ intersect at one point or are parallel.

Artemchuk O., Moroz M.

2020 Ukraine MO grade X P3 Online

Let $H_a,H_b,H_c$ be feet of altitudes from the corresponding vertices of the triangle $ABC, H$ be the orthocenter of this triangle, and $K$ is a point that is symmetric to$H$ wrt $BC$. A line passing through point $H$ parallel to $H_bH_c$, intersects line $AB$ and $AC$ at points $X$ and $Y$ respectively. . Prove that the circles are circumscribed to $\vartriangle ABC$ and $\vartriangle XYK$ are tangent.

Bondarenko Mykhailo

2020 Ukraine MO grade X P6  XI P5 Online

An isosceles triangle $ABC$ is given with a base $AC, P$ is an arbitrary point on this base, $T$ is the projection of $P$ on $BC$. In what ratio does the symmedian of $\vartriangle PBC$ derived from vertex $C$, divides the segment $AT$?

Anton Trygub

2020 Ukraine MO grade XI P4 Online

Let $ABC$ be an acute not isosceles triangle. Its bisectors $AL_1$ and $BL_2$ intersect at a point $I$ . Points $D$ and $E$ are selected on segments $AL_1$ and $BL_2$ respevticely in such a way that $\angle DBC =\frac12 \angle A$ and $\angle EAC =\frac12 \angle B$.  Lines $AE$ and $BD$ intersect at a point $P$. Point $K$ is symmetric to the point $I$ wrt $DE$. Prove that lines $KP$ and $DE$ intersect on the circumscribed circle of $\vartriangle ABC$.

Hilko Danilo

2021 Ukraine MO grade VIII P4

Triangle $ABC$ is isosceles with a vertex at point $A$. Inside $\vartriangle ABC$ points $P$ and $Q$ are selected such that $\angle BPC = \frac{3}{2}\angle BAC$, $BP=AQ$  and $AP=CQ$. Prove that $AP=PQ$.

Fedir Yudin

2021 Ukraine MO grade IX P4

Given a triangle $ABC$ in which $\angle A= 60^o$. On the sides $AB$ and $AC$ mark points $P$ and $Q$, respectively, that  $BP=PQ=QC$. Prove that the circumcircle of the triangle $APQ$  passes through the projection of the orthocenter of the triangle $ABC$ on its median respective to the side $BC$.

Fedir Yudin

2021 Ukraine MO grade IX P6 X P6

Circles $w_1$ and $w_2$ intersect at points $P$ and $Q$ and touch a circle $w$ with center at point $O$ internally at points $A$ and $B$, respectively. It is known that the points $A,B$ and $Q$ lie on one line. Prove that the point $O$ lies on the external bisector $\angle APB$. 

Nazar Serdyuk 

2021 Ukraine MO grade X P4 , XI P4

Let $O, I, H$ be the circumcenter, the incenter, and the orthocenter of $\triangle ABC$. The lines $AI$ and $AH$ intersect the circumcircle of $\triangle ABC$ for the second time at $D$ and $E$, respectively. Prove that if $OI \parallel BC$, then the circumcenter of $\triangle OIH$ lies on $DE$.

Fedir Yudin

2021 Ukraine MO grade XI P6

The altitudes $AA_1, BB_1$ and $CC_1$ were drawn in the triangle $ABC$. Point $K$ is a projection of point $B$ on $A_1C_1$. Prove that the symmmedian $\vartriangle ABC$ from the vertex $B$ divides the segment $B_1K$ in half.

Anton Trygub

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http://ri.kharkov.ua/olympiad/old.html

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