Kazakhstan 1999 -2019 IX-XI 113p

geometry problems from Kazakhstan Mathematical Olympiads
with aops links in the names
collected inside aops here: 

1999 - 2019


1999 Kazakhstan MO grade IX P4
Given a rectangle $ ABCD $ with a larger side $ AB $. The circle centered at $ B $ with radius $ AB $ intersects the line $ CD $ at points $ E $ and $ F $. Prove that:
a) the circle circumscribed around the triangle $ EBF $ it tangent to a circle with a diameter of $ AD $.
b) If $ G $ is the intersection point of these circles, then the points $ D $, $ G $, $ B $ lie on one straight line.

1999 Kazakhstan MO grade IX P6
The diagonals of the trapezoid $ ABCD $ are mutually perpendicular ($ AB $ is a larger base). Let $ O $ be the center of the circumcircle of triangle $ ABC $, and $ E $ be the intersection point of $ OB $ and $ CD $. Prove that $ BC ^ 2 = CD \ cdot CE $.

1999 Kazakhstan MO grade X P2
One square is obtained by rotating the second square relative to its center by the angle $ \alpha $  $(\alpha \leq \pi / 4) $. At what value of $ \alpha $, the perimeter of the octagon, the common part of the two squares, has the minimum value.

1999 Kazakhstan MO grade X P8
In the acute isosceles triangle $ ABC $, with base $ AC $, altitudes $ AA_1 $ and $ BB_1 $ are drawn. The straight line passing through $ B $ and the midpoint of $ AA_1 $ intersects the circle $ \omega $ circumscribed around the triangle $ ABC $ at the point $ E $. The tangent to $ \omega $ at the point $ A $ intersects the line $ BB_1 $ at the point $ D $. Prove that the points $ D $, $ E $, $ B_1 $ and $ C $ lie on the same circle.

1999 Kazakhstan MO grade XI P3
The circle inscribed in the triangle $ ABC $ touches the sides $ AB $ and $ BC $ at the points $ C_1 $ and $ A_1 $, respectively. The lines $ CO $ and $ AO $ intersect the line $ C_1A_1 $ at the points $ K $ and $ L $. $ M $ is the middle of $ AC $ and $ \angle ABC = 60^\circ $. Prove that $ KLM $ is a regular triangle.

On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $

2000 Kazakhstan MO grade IX P1
Given a quadrangle $ PQRS $ around which a circle can be circumscribed and $ \angle PSR = 90^\circ $. $ H $ and $ K $ are the bases of perpendiculars dropped from the point $ Q $ on the lines $ PR $ and $ PS $ respectively. Prove that the line $ HK $ divides the segment $ QS $ in half.

2000 Kazakhstan MO grade IX P7
A circle is described around the triangle $ ABC $. $ A '$, $ B' $, $ C '$ respectively the midpoints of $ BC, CA, AB $. The sides $ BC, CA $, and $ AB $ intersect pairs of segments $ (C'A ', A'B') $, $ (A'B ', B'C') $ and $ (B'C ', C 'A') $ in pairs of points $ (M, N) $, $ (P, Q) $ and $ (R, S) $ respectively. Prove that $ MN = PQ = RS $ if and only if the triangle $ ABC $ is equilateral.

2000 Kazakhstan MO grade X P3
Let the point $ O $ be the center of the circle. Two equal chords $ AB $ and $ CD $ intersect at $ L $ in such a way that $ AL> LB $ and $ DL> LC $. Let $ M $ and $ N $, respectively, be points on segments $ AL $ and $ DL $ such that $ \angle ALC = 2 \angle MON $. Prove that the chord of a circle passing through the points $ M $ and $ N $ is equal to $ AB $ and $ CD $.

2000 Kazakhstan MO grade X P7, XI P8
Given a triangle $ ABC $ and a point $ M $ inside it. Prove that $
\min \{MA, MB, MC\} + MA + MB + MC <AB + BC + AC. $

Given a circle centered at $ O $ and two points $ A $ and $ B $ lying on it. $ A $ and $ B $ do not form a diameter. The point $ C $ is chosen on the circle so that the line $ AC $ divides the segment $ OB $ in half. Let lines $ AB $ and $ OC $ intersect at $ D $, and let lines $ BC $ and $ AO $ intersect  at $ F $. Prove that $ AF = CD $.

2001 Kazakhstan MO grade IX P2
Let $ M $ be the midpoint of side $ BC $ of a triangle $ ABC $. On the line $ AC $, there are distinct points $ L $ and $ N $ such that $ AL = CN $ and $ CL = AN $. Prove that the lines $ LM $ and $ MN $ at the intersection with the line $ AB $ form equidistant points with respect to $ A $ and $ B $, respectively.

2001 Kazakhstan MO grade IX P5
Given two circles that have at least one common point. A point $ M $ is called  singular   if two different lines $ l $ and $ m $ passing through $ M $ and forming $ A $ and $ B $ at the intersection of line $ l $ with the first circle, and the intersection of the line $ m $ with the second circle points $ C $ and $ D $ such that the resulting four points lie on the same circle. Find the locus of all  singular  points.

2001 Kazakhstan MO grade X P3
In the circle with center $ O $ there is inscribed a quadrilateral $ ABCD $, different from trapezium. Let $ M $ be the intersection point of the diagonals, $ K $ be the intersection point of the circles described around the triangles $ BMC $ and $ DMA $, $ L $ be the intersection point of the circles circumscribed around the triangles $ AMB $ and $ CMD $, where $ K $, $ L $ and $ M $ are different points. Prove that around the $ OLMK $ quadrilateral a circle can be circumscribed.

2001 Kazakhstan MO grade X P6
Let a point $ M $, distinct from the middle of $ AC $, be fixed on the line $ AC $ of the triangle $ ABC $. For any point $ K $ line $ BM $, other than $ B $ and $ M $, a line $ LN $ is constructed such that $ L $ is the intersection point of $ AK $ and $ BC $, and $ N $ is the intersection point of $ CK $ and $ AB $. Prove that all such lines $ LN $ intersect at one point.

2001 Kazakhstan MO grade XI P2
In the acute triangle $ ABC $, $ L $, $ H $ and $ M $ are the intersection points of bisectors, altitudes and medians, respectively, and $ O $ is the center of the circumscribed circle. Denote by $ X $, $ Y $ and $ Z $ the intersection points of $ AL $, $ BL $ and $ CL $ with a circle, respectively. Let $ N $ be a point on the line $ OL $ such that the lines $ MN $ and $ HL $ are parallel. Prove that $ N $ is the intersection point of the medians of $ XYZ $.

2001 Kazakhstan MO grade XI P7
Two circles $ w_1 $ and $ w_2 $ intersect at two points $ P $ and $ Q $. The common tangent to $ w_1 $ and $ w_2 $, which is closer to the point $ P $ than to $ Q $, touches these circles at $ A $ and $ B $, respectively. The tangent to $ w_1 $ at the point $ P $ intersects $ w_2 $ at the point $ E $ (different from $ P $), and the tangent to $ w_2 $ at the point $ P $ intersects $ w_1 $ at $ F $ (different from $ P $). Let $ H $ and $ K $ be points on the rays $ AF $ and $ BE $, respectively, such that $ AH = AP $ and $ BK = BP $. Prove that the points $ A $, $ H $, $ Q $, $ K $ and $ B $ lie on the same circle.

2002 Kazakhstan MO grade IX P1
A square $ ABCD $ with side $1$ is given. On the sides $ BC $ and $ CD $, points $ M $ and $ N $ are chosen, respectively, so that the perimeter of the triangle $ MCN $ is 2. Find the distance from $ A $ to $ MN $.

2002 Kazakhstan MO grade IX P6
In the triangle $ ABC $ $ \angle B> 90^{\circ} $ and on the side of $ AC $ for some point $ H $ $ AH = BH $ with the line $ BH $ perpendicular to $ BC $. Denote by $ D $ and $ E $ the midpoints of $ AB $ and $ BC $, respectively. The line drawn through $ H $ and parallel to $ AB $ intersects $ DE $ at $ F $. Prove that $ \angle BCF = \angle ACD. $

2002 Kazakhstan MO grade X P3
Find the smallest number $ c $ that satisfies the following property: on the sides of any triangle with perimeter $1$, you can find two points dividing the perimeter in half and spaced at a distance of no more than $ c $.

2002 Kazakhstan MO grade X P6 , XI P5
On the plane is given the acute triangle $ ABC $. Let $ A_1 $ and $ B_1 $ be the bases of the altitudes of $ A $ and $ B $ drawn from those vertices, respectively. Tangents at points $ A_1 $ and $ B_1 $ drawn to the circumscribed circle of the triangle $ CA_1B_1 $ intersect at $ M $. Prove that the circles circumscribed around the triangles $ AMB_1 $, $ BMA_1 $ and $ CA_1B_1 $ have a common point.

Let $ O $ be the center of the inscribed circle of the triangle $ ABC $, tangent to the side of $ BC $. Let $ M $ be the midpoint of $ AC $, and $ P $ be the intersection point of $ MO $ and $ BC $. Prove that $ AB = BP $ if $ \angle BAC = 2 \angle ACB $.

2003 Kazakhstan MO grade IX P3
In $ \triangle ABC $ it is known that $ \ angle C> 10^{\circ} $ and $ \angle B = \angle C + 10^{\circ} $. Consider the points $ E, D $ on the segments $ AB $ and $ AC $, respectively, such that $ \angle ACE = 10^{\circ} $ and $ \angle ABD = 15^{\circ} $. Let a point $ Z $, other than $ A $, be the intersection of the circumscribed circles of triangles $ ABD $ and $ AEC $. Prove that $ \angle ZBA> \angle ZCA $.

2003 Kazakhstan MO grade IX P6
In an acute triangle the points $ D $ and $ E $ are the bases of the altitudes from the vertices $ A $ and $ B $, respectively, $ AC> BC $ and $ AB = 2DE $. Denote by $ O $ and $ I $ respectively the centers of the circumscribed and inscribed circles of the triangle. Find the angle $ \angle AIO $.
         
2003 Kazakhstan MO grade X P2
Let the points $ M $ and $ N $ in the acute triangle $ ABC $ be interior points of the sides $ AC $ and $ BC $ respectively, and $ K $ be the midpoint of the segment $ MN $. $ D $ is the intersection point of the circumscribed circles of the triangles $ CAN $ and $ BCM $ different from the point $ C $. Prove that the line $ CD $ passes through the center of the circumscribed circle of the triangle $ ABC $ if and only if the perpendicular bisector of the segment $ AB $ passes through the point $ K $.

2003 Kazakhstan MO grade X P5
Given a triangle $ ABC $ with acute angles $ B $ and $ C $. The rectangle $ KLMN $ is inscribed in it so that the points $ L $ and $ M $ lie on the sides $ AB $ and $ AC $, respectively, and the points $ N $ and $ K $ are on the side $ CB $. The point $ O $ is the center of the rectangle. The lines $ BO $ and $ CO $ intersect the sides of the rectangle $ MN $ and $ LK $ at the points $ C_1 $ and $ B_1 $, respectively. Prove that the lines $ AO, BB_1 $ and $ CC_1 $ intersect at one point.

2003 Kazakhstan MO grade XI P4
Let the inscribed circle $ \omega $ of triangle $ ABC $ touch the side $ BC $ at the point $ A '$. Let $ AA '$ intersect $ \omega $ at $ P \neq A $. Let $ CP $ and $ BP $ intersect $ \omega $, respectively, at points $ N $ and $ M $ other than $ P $. Prove that $ AA ', BN $ and $ CM $ intersect at one point.

2003 Kazakhstan MO grade XI P6
Let the point $ B $ lie on the circle $ S_1 $ and let the point $ A $, other than the point $ B $, lie on the tangent to the circle $ S_1 $ passing through the point $ B $. Let a point $ C $ be chosen outside the circle $ S_1 $, so that the segment $ AC $ intersects $ S_1 $ at two different points. Let the circle $ S_2 $ touch the line $ AC $ at the point $ C $ and the circle $ S_1 $ at the point $ D $, on the opposite side from the point $ B $ with respect to the line $ AC $. Prove that the center of the circumcircle of triangle $ BCD $ lies on the circumcircle of triangle $ ABC $.

2004 Kazakhstan MO grade IX P3
In the acute triangle $ ABC $, the point $ D $ is the base of the altitude from the vertex $ C $, and $ M $ is the midpoint of the side $ AB $. The straight line passing through $ M $ intersects the rays $ CA $ and $ CB $ respectively at the points $ K $ and $ L $ so that $ CK = CL $. Let $ S $ be the center of the circumscribed circle of the triangle $ CKL $. Prove that $ SD = SM $.

2004 Kazakhstan MO grade IX P6
The acute triangle $ ABC $, where $ \angle ABC = 2 \angle ACB $, is sicrumscribed by a circle with center $ O $. Let $ K $ be the intersection point of $ AO $ and $ BC $, and point $ O_1 $ be the center of the circumscribed circle of triangle $ ACK $. Prove that the area of the quadrilateral $ AKCO_1 $ is equal to the area of the triangle $ ABC $.

2004 Kazakhstan MO grade X P5
In the triangle $ ABC $ the side $ BC $ is the smallest. On the rays $ BA $ and $ CA $, the segments $ BD $ and $ CE $ are equal to $ BC $. Prove that the radius of the circumcircle of the triangle $ ADE $ is $ \sqrt{R ^ 2 - 2Rr} $ (where $ R $ and $ r $ are the radii of the circumscribed and inscribed circles of the triangle $ ABC $).

Let $ ABCD $ be a convex quadrilateral with $ AB $ not parallel to $ CD $, and let $ X $ be a point inside $ ABCD $ such that $ \angle ADX = \angle BCX <90^\circ $ and $ \angle DAX = \angle CBX < 90^\circ $. If $ Y $ is the intersection point of the midperpendiculars $ AB $ and $ CD $, then prove that $ \angle AYB = 2 \angle ADX $..

2005 Kazakhstan MO grade IX P3
Let $ M $ be the intersection point of the segments $ AL $ and $ CK $, where the points $ K $ and $ L $ lie respectively on the sides $ AB $ and $ BC $ of the triangle $ ABC $ so that the quadrilaterals $ AKLC $ and $ KBLM $ are cyclic. Find the angle $ \angle ABC $ if the radii of the circles circumscibed around those cyclic quadrilaterals are equal.

2005 Kazakhstan MO grade IX P5
On the $ CD $ side of the trapezoid $ ABCD $ ($ BC \parallel AD $), the point $ K $ is marked so that the triangle $ ABK $ is equilateral. Prove that on the line $ AB $ there exists a point $ L $ such that the triangle $ CDL $ is also equilateral.

In the acute triangle $ ABC $, the angle $ \angle A = 45^\circ $, and the heights $ BB_1 $ and $ CC_1 $ intersect at $ H $. Prove that the lines $ BC $, $ B_1C_1 $ and the line $ l $ passing through $ A $ perpendicular on $ AC $, intersect at one point if and only if $ H $ is the midpoint of the segment $ BB_1 $.

2005 Kazakhstan MO grade XI P6
The line parallel to side $AC$ of a right triangle $ABC$ $(\angle C=90^\circ)$ intersects sides $AB$ and $BC$ at $M$ and $N$, respectively, so that the $CN / BN = AC / BC = 2$. Let $O$ be the intersection point of the segments $AN$ and $CM$ and $K$ be a point on the segment $ON$ such that $MO + OK = KN$. The perpendicular line to  $AN$ at point $K$ and the bisector of triangle $ABC$ of $\angle B$ meet at point $T$. Find the angle $\angle MTB$.

2006 Kazakhstan MO grade IX P3 , X P4, XI P4
grade IX P4, X P3
The bisectors of the angles $ A $ and $ C $ of the triangle $ ABC $ intersect the circumscirbed circle of this triangle at the points $ A_0 $ and $ C_0 $, respectively. The straight line passing through the center of the inscribed circle of triangle $ ABC $ parallel to the side of $ AC $, intersects with the line $ A_0C_0 $ at $ P $. Prove that the line $ PB $ is tangent to the circumcircle of the triangle $ ABC $.

grade XI P4
The bisectors of the angles $ A $ and $ C $ of the triangle $ ABC $ intersect the sides at the points $ A_1 $ and $ C_1 $, and the circumcircle of this triangle at points $ A_0 $ and $ C_0 $ respectively. Straight lines $ A_1C_1 $ and $ A_0C_0 $ intersect at point $ P $. Prove that the segment connecting $ P $ with the center inscribed circles of triangle $ ABC $, parallel to $ AC $.

2006 Kazakhstan MO grade IX P6
In the acute triangle $ ABC $, a bisector of $ AD $ is drawn and the altitude $ BE$.
Prove that the angle $\angle CED $ is more than $ 45^\circ $.

2006 Kazakhstan MO grade X P6
Through the intersection point of the heights of the acute triangle $ ABC $ there are three circles, each of which touches one of the side of the triangle at the base of altitude. Prove that the second intersection points circles are vertices of a triangle, similar to the original.

In the tetrahedron $ ABCD $ from the vertex $ A $, the perpendiculars $ AB '$, $ AC' $ are drawn, $ AD '$ on planes dividing dihedral angles at edges $ CD $, $ BD $, $ BC $ in half. Prove that the plane $ (B'C'D ') $ is parallel to the plane $ (BCD) $.

2007 Kazakhstan MO grade IX P2
Given the triangle $ ABC $. The point $ R $ is chosen on the extension of the side $ AB $ for the point $ B $ so that $ BR = BC $, and the point $ S $ is chosen on the extension of the side $ AC $ for the point $ C $ so that $ CS = CB $ . The diagonals of the quadrilateral $ BRSC $ intersect at $ A '$. The points $ B '$ and $ C' $ are defined similarly. Prove that the area of the hexagon $ AC'BA'CB '$ is equal to the sum of the areas of the triangles $ ABC $ and $ A'B'C' $.

2007 Kazakhstan MO grade IX P5
Let $ I $ be the center of a circle inscribed in a triangle $ ABC $, $ BP $ be the bisector of the angle $ \angle ABC $, $ P $ lies on $ AC $. Prove that if $ AP + AB = CB $, then the triangle $ API $ is isosceles.

2007 Kazakhstan MO grade X P2
The acute triangle $ ABC $ is inscribed in a circle with the center $ O $. The point $ P $ is chosen on the smaller of the two arcs $ AB $. The line passing through $ P $ perpendicular on $ BO $ intersects the sides $ AB $ and $ BC $ at the points $ S $ and $ T $, respectively. The line passing through $ P $ perpendicular on $ AO $ intersects the sides $ AB $ and $ AC $ at points $ Q $ and $ R $, respectively. Prove that:
a) the triangle $ PQS $ is isosceles;
b) $ PQ ^ 2 = QR \cdot ST. $

2007 Kazakhstan MO grade X P5
In the triangle $ ABC $, the point $ M $ is the midpoint of the side $ AB $, $ BD $ is the bisector of the angle $ \angle ABC $, $ D $ lies on $ AC $. It is known that $ \angle BDM = 90^ \circ $. Find the ratio $ AB: BC $.

2007 Kazakhstan MO grade XI P2
Let $ABC$ be an isosceles triangle with $AC = BC$ and $I$ is the center of the inscribed circle. The point $P$ lies on the circle circumscribed about the triangle $AIB$ and lies inside the triangle $ABC$. Straight lines passing through point $P$ parallel to $CA$ and $CB$ intersect $AB$ at points $D$ and $E$, respectively. The line through $P$ which is parallel to $AB$ intersects $CA$ and $CB$ at points $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ meet at the circumcircle of triangle $ABC$.

2007 Kazakhstan MO grade XI P5
Convex quadrilateral $ABCD$ with $AB$ not equal to $DC$ is inscribed in a circle. Let $AKDL$ and $CMBN$ be rhombs with same side of $a$. Prove that the points $K, L, M, N$ lie on a circle.

2008 Kazakhstan MO grade IX P2
An circle with center $ I_a $ is tangent to the side $ BC $ and extensions of the sides $ AC $ and $ AB $ of the triangle $ ABC $. We denote by $ B_1 $ the middle of the arc $ AC $ of the circumcircle of the triangle $ ABC $ containing the vertex $ B $. Prove that the points $ I_a $ and $ A $ are equidistant from the point $ B_1 $.

2008 Kazakhstan MO grade IX P4
The cyclic quadrilateral $ ABCD $ is given. Let the extensions of the sides $ AB $ and $ CD $ beyond points $ B $ and $ C $, respectively, intersect at the point $ M $. We denote the feet of the perpendiculars from the point $ M $ on the diagonals $ AC $ and $ BD $ by $ P $ and $ Q $, respectively. Prove that $ KP = KQ $ where, $ K $ is the middle of the side $ AD $.

2008 Kazakhstan MO grade X P2
The bisector of the angle $ A $ of the triangle $ ABC $ intersects the side $ BC $ at the point $ A_1 $, and the circumcircle at $ A_0 $. The points $ C_1 $ and $ C_0 $ are defined similarly. The lines $ A_0C_0 $ and $ A_1C_1 $ intersect at the point $ P $. Prove that $ PI $ is parallel to the side $ AC $, where $ I $ is the center of the inscribed circle.

2008 Kazakhstan MO grade X P4
Two circles are given tangent internally at the point $ N $. The chords $ BA $ and $ BC $ of the outer circle touch the interior circle at the points $ K $ and $ M $, respectively. Let $ Q $ and $ P $ be, respectively, the midpoints of the arcs $ AB $ and $ BC $ that do not contain the point $ N $. The circumcircles of the triangles $ BQK $ and $ BPM $ intersect at the point $ B_1 \neq B $. Prove that $ BPB_1Q $ is a parallelogram.

2008 Kazakhstan MO grade XI P2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.

2008 Kazakhstan MO grade XI P5
Let $ \triangle ABC$ be a triangle and let $ K$ be some point on the side $ AB$, so that the tangent line from $ K$ to the incircle of $ \triangle ABC$ intersects the ray $ AC$ at $ L$. Assume that $ \omega$ is tangent to sides $ AB$ and $ AC$, and to the circumcircle of $ \triangle AKL$. Prove that $ \omega$ is tangent to the circumcircle of $ \triangle ABC$ as well.

2009 Kazakhstan MO grade IX P1
In the triangle $ABC$ the inscribed circle touches sides $BC$, $CA$ and $AB$ , at  points $A_1$, $B_1$ and  $C_1$  respectively. Denote the orthocenters of the triangles $AC_1B_1$ and  $CA_1B_1$ as $H_1$ and  $H_2$ . Prove that the quadrilateral $AH_1H_2C$  is cyclic.

2009 Kazakhstan MO grade IX P5
Given triangle $ABC$, where $AB>AC$.Altitudes $CC_1$ and $BB_1$ intersect at $H$, lines $B_1C_1$ and BC at $P$. Let $M$ be midpoint of $BC$, line segments $MH$ & $AP$ intersect each other at $K$ respectively. Prove that $KM$ is angle bisector of $B_1KB$.

2009 Kazakhstan MO grade X P2
Let incircle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively.
Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be incenter and circumcenter of $ABC$ respectively. Prove, that $I, O, H$ lies on one line.

2009 Kazakhstan MO grade X P5, grade XI P5
Quadrilateral $ABCD$ inscribed in circle with center $O$. Let lines $AD$ and $BC$ intersects at $M$, lines $AB$ and $CD$- at $N$, lines $AC$ and $BD$ -at $P$, lines $OP$ and $MN$ at $K$.
Proved that $ \angle AKP = \angle PKC$.

2009 Kazakhstan MO grade XI P2
In triangle $ABC$ $AA_1; BB_1; CC_1$-altitudes. Let $I_1$ and $I_2$ be incenters of triangles $AC_1B_1$ and $CA_1B_1$ respectively. Let incircle of $ABC$ touch $AC$ in $B_2$.
Prove, that quadrilateral $I_1I_2B_1B_2$ inscribed in a circle
.
2010 Kazakhstan MO grade IX P1
Triangle $ABC$ is given. Circle $ \omega $ passes through $B$, touch $AC$ in $D$ and intersect sides $AB$ and $BC$ at $P$ and $Q$ respectively. Line $PQ$ intersect $BD$ and $AC$ at $M$ and $N$ respectively. Prove that $ \omega $, circumcircle of $DMN$ and circle, touching $PQ$ in $M$ and passes through B, intersects in one point.

2010 Kazakhstan MO grade IX P5
Arbitrary triangle $ABC$ is given (with $AB<BC$). Let $M$ - midpoint of $AC$, $N$- midpoint of arc $AC$ of circumcircle $ABC$, which is contains point $B$. Let $I$ - incenter of $ABC$. Proved, that $ \angle IMA = \angle INB$

2010 Kazakhstan MO grade X P1
Triangle $ABC$ is given. Consider ellipse $ \Omega _1$, passes through $C$ with focuses in $A$ and $B$. Similarly define ellipses $ \Omega _2 , \Omega _3$ with focuses $B,C$ and $C,A$ respectively. Prove, that if all ellipses have common point $D$ then $A,B,C,D$ lies on the circle.

(Ellipse with focuses $X,Y$, passes through $Z$- locus of point $T$, such that $XT+YT=XZ+YZ$)

2010 Kazakhstan MO grade X P5
On sides of convex quadrilateral $ABCD$ on external side constructed equilateral triangles $ABK, BCL, CDM, DAN$. Let $P,Q$- midpoints of $BL, AN$ respectively and $X$- circumcenter of $CMD$. Prove, that $PQ$ perpendicular to $KX$

2010 Kazakhstan MO grade XI P3
Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$. Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$

2010 Kazakhstan MO grade XI P5
Let $O$ be the circumcircle of acute triangle $ABC$, $AD$-altitude of $ABC$ ($ D \in BC$), $ AD \cap CO =E$, $M$-midpoint of $AE$, $F$-feet of perpendicular from $C$ to  $AO$.
Proved that point of intersection $OM$ and $BC$ lies on circumcircle of triangle $BOF$

2011 Kazakhstan MO grade IX P1
The quadrilateral $ABCD$ is circumscribed about the circle, touches the sides $AB, BC, CD, DA$ in the points $K, L, M, N,$ respectively. Let $P, Q, R, S$ midpoints of the sides $KL, LM, MN, NK$. Prove that $PR = QS$ if and only if $ABCD$ is inscribed.

2011 Kazakhstan MO grade IX P5, grade X P5
Given a non-degenerate triangle $ABC$, let $A_{1}, B_{1}, C_{1}$ be the point of tangency of the incircle with the sides $BC, AC, AB$. Let $Q$ and $L$ be the intersection of the segment $AA_{1}$ with the incircle and the segment $B_{1}C_{1}$ respectively. Let $M$ be the midpoint of $B_{1}C_{1}$. Let $T$ be the point of intersection of $BC$ and $B_{1}C_{1}$. Let $P$ be the foot of the perpendicular from the point $L$ on the line $AT$. Prove that the points $A_{1}, M, Q, P$ lie on a circle.

2011 Kazakhstan MO grade X P1
Inscribed in a triangle $ABC$ with the center of the circle $I$ touch the sides $AB$ and $AC$ at points $C_{1}$ and $B_{1}$, respectively. The point $M$ divides the segment $C_{1}B_{1}$ in a 3:1 ratio, measured from $C_{1}$. $N$ - the midpoint of $AC$. Prove that the points $I, M, B_{1}, N$ lie on a circle, if you know that $AC = 3 (BC-AB)$.

2011 Kazakhstan MO grade XI P2
Let $w$-circumcircle of triangle $ABC$ with an obtuse angle $C$ and $C '$symmetric point of point $C$ with respect to $AB$. $M$ midpoint of $AB$. $C'M$ intersects $w$ at $N$ ($C '$ between $M$ and $N$). Let $BC'$ second crossing point $w$ in $F$, and $AC'$ again crosses the $w$ at point $E$. $K$-midpoint $EF$. Prove that the lines $AB, CN$ and$ KC'$ are concurrent.

2012 Kazakhstan MO grade IX P2
Given two circles $k_{1}$ and $k_{2}$ with centers $O_{1}$ and $O_{2}$ that intersect at the points $A$ and $B$.Passes through A two lines that intersect the circle $k_{1}$ at the points $N_{1}$and $M_{1}$, and the circle $k_{2}$ at the points $N_{2}$ and $M_{2}$ (points $A, N_{1},M_{1}$ in colinear). Denote the midpoints of the segments $N_{1}N_{2}$ and $M_{1}M_{2]}$ , through $N$ and $M$.Prove that:
a) Points $M,N,A$ and $B$ lie on a circle
b) The center of the circle passing through $M,N,A$ and $B$ lies in the middle of the segment $O_{1}O_{2}$

2012 Kazakhstan MO grade IX P5
Given an inscribed quadrilateral $ABCD$, which marked the midpoints of the points $M, N, P, Q$ in this order. Let diagonals $AC$ and $BD$ intersect at point $O$. Prove that the triangle $OMN, ONP, OPQ, OQM$ have the same radius of the circles

2012 Kazakhstan MO grade X P2
Let $ABCD$ be an inscribed quadrilateral, in which $\angle BAD<90$. On the rays $AB$ and $AD$ are selected points $K$ and $L$, respectively, such that$ KA = KD, LA = LB$. Let $N$ - the midpoint of $AC$.Prove that if $\angle BNC=\angle DNC $,so $\angle KNL=\angle BCD $

2012 Kazakhstan MO grade X P4
Let $k_{1},k_{2}, k_{3}$ -Excircles triangle $A_{1}A_{2}A_{3}$ with area $S$. $ k_{1}$ touch side $A_{2}A_{3} $ at the point $B_{1}$ Direct $A_{1}B_{1}$ intersect $k_{1}$ at the points $B_{1}$ and $C_{1}$.Let $S_{1}$ - area of ​​the quadrilateral $A_{1}A_{2}C_{1}A_{3}$ Similarly, we define $S_{2}, S_{3}$. Prove that $\frac{1}{S}\le \frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{2}}$

2012 Kazakhstan MO grade XI P3
Line $PQ$ is tangent to the incircle of triangle $ABC$ in such a way that the points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively. On the sides $AB$ and $AC$ are selected points $M$ and $N$, respectively, so that $AM = BP$ and $AN = CQ$. Prove that all lines constructed in this manner $MN$ pass through one point

2012 Kazakhstan MO grade XI P5
Given the rays $ OP$ and $OQ$.Inside the smaller angle $POQ$ selected points $M$ and $N$, such that $\angle POM=\angle QON $ and $\angle POM<\angle PON $ The circle, which concern the rays $OP$ and $ON$, intersects the second circle, which concern the rays $OM$ and $OQ$ at the points $B$ and $C$. Prove that$\angle POC=\angle QOB $

2013 Kazakhstan MO grade IX P3
Given a triangle $ABC$, about which describes a circle with center $O$. Let $I$ be the center of the inscribed circle of triangle $ABC$, and the point $A_1  (A_1\neq A
)$ and $B_1  (B_1 \neq B)$ on circumscribed circle such that the angle $\angle IA_1B = \angle IA _1C$ and $\angle IB_1A = \angle IB_1C$. Prove that lines $AA_1$ and $BB_1$ intersect on the line $OI$.

2013 Kazakhstan MO grade IX P5
Let $AD, BE$ and $CF$ bisector of triangle $ABC$. Denoted by $M$ and $N$ are the midpoints of $DE$ and $DF$, respectively. Prove that if $ \angle BAC \geq 60$, then $BN + CM <BC$

2013 Kazakhstan MO grade X P3
Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$, and let $AB$ and $CD$ intersect at $K$. Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$. Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$. Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.

2013 Kazakhstan MO grade X P5, grade XI P5
Let in triangle $ABC$ incircle touches sides $AB,BC,CA$ at $C_1,A_1,B_1$ respectively. Let $\frac {2}{CA_1}=\frac {1}{BC_1}+\frac {1}{AC_1}$ .Prove that if $X$ is intersection of incircle and $CC_1$ then $3CX=CC_1$

2013 Kazakhstan MO grade XI P1
Given triangle $ABC$ with incenter $I$. Let $P,Q$ be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

2014 Kazakhstan MO grade IX P1
In a triangle $ABC$ a point $I$ — the inscribed circle center, and $w$ — described circle. Lines $BI$ and $CI$ cross $w$ respectively in points $B_1$ and $C_1$, and line $B_1C_1$ intersect lines $AB$ and $AC$ in points $C_2$ and $B_2$, respectively. Let $w_1$— triangle circumscribed circle $IB_1C_1$ and lines $IB_2$ and $IC_2$ cross $w_1$ 1 in points $M$ and $N$, respectively. Prove that $BC_2*B_2C = B_2M*C_2N$

2014 Kazakhstan MO grade IX P5
In a convex quadrilateral $ABCD$ the following ratios are fairs:
$AB = BC, AD = BD$ and $\angle ADB = 2 \angle BDC$. It is known that $\angle ACD = 100 $. Find $\angle ADC.$

2014 Kazakhstan MO grade X P3
The triangle $ABC$ is inscribed in a circle $w_1$. Inscribed in a triangle circle touchs the sides $BC$ in a point $N$. $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$, and passing through a point $N$. Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$) respectively. Prove, that lines $AO$ and $JN$ are parallel.

2014 Kazakhstan MO grade X P5
About not isosceles triangle $ABC$ the circle $w$, a point $M$ is described —middle $AC$. The tangent to $w$ in a point $B$ crosses a straight line $AC$ in a point $N$, and a straight line $BM$ repeatedly crosses $w$ in a point $L$. Let the point $P$ be symmetric to a point $L$ relatively $M$. The circle, circumscribed about a triangle $BPN$, repeatedly crosses line $AN$ in a point $Q$. Prove that $\angle ABP = \angle QBC.$

2014 Kazakhstan MO grade XI P3
The triangle $ABC$ is inscribed in a circle $w_1$. Inscribed in a triangle circle touchs the sides $BC$ in a point $N$. $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$, and passing through a point $N$. Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$) respectively. Prove, that lines $AO$ and $JN$ are parallel.

2014 Kazakhstan MO grade XI P4
Given a scalene triangle $ABC$. Incircle of $\triangle{ABC{}}$ touches the sides $AB$ and $BC$ at points $C_1$ and $A_1$ respectively, and excircle of $\triangle{ABC}$ (on side $AC$) touches $AB$ and $BC$ at points $ C_2$ and $A_2$ respectively. $BN$ is bisector of $\angle{ABC}$ ($N$ lies on $BC$). Lines $A_1C_1$ and $A_2C_2$ intersects the line $AC$ at points $K_1$ and $K_2$ respectively. Let circumcircles of $\triangle{BK_1N}$ and $\triangle{BK_2N}$ intersect circumcircle of a $\triangle{ABC}$ at points $P_1$ and $P_2$ respectively. Prove that $AP_1$=$CP_2$

2015 Kazakhstan MO grade IX P3
Given right trangle with $\angle{C}=90^{\circ}$. Inscribed and escribed circles of $ABC$ are tangent to side $BC$ at points $A_1$ and $A_2$. Similarly, we define the points $B_1$ and $B_2$. Prove that the segments $A_1B_2$ and $B_1A_2$ intersect at an altitude drawn from vertex $C$ of the triangle $ABC$.

2015 Kazakhstan MO grade IX P4
In triangle $ABC$, point $D$ - base bisector of $B$, and the point $M$ - the midpoint of side $AC$. On the segment $BD$ there are points $A_1$ and $C_1$ such that $DA = DA_1$ and $DC = DC_1$. Lines $AA_1$ and $CC_1$ meet at $E$. Line $ME$ intersects $BC$ at $F$. Prove equality $AB + BF = CF$.

2015 Kazakhstan MO grade X P1
Circle $W$, described about the triangle $ABC$, intersects the sides $AD$ and $DC$ of the parallelogram $ABCD$, for the second time in points $A_1$ and $C_1$ respectively. Let $E$ be the intersection point of lines $AC$ and $A_1C_1$. Let $BF$ - diameter $W$, and the point $O_1$ is symmetric to the center of $W$ with respect to $AC$. Prove that the lines $FO_1$ and $DE$ are perpendicular.

2015 Kazakhstan MO grade X P5
Given two circles $W_1$ and $W_2$, the segments $AB$ and $CD$ - the common external tangents to them (points $A$ and $C$ lie on $W_1$, and the points $B$ and $D$ - on the $W_2$). Line $AD$ second time intersects the circle $W_1$ at the point $P$, and the circle $W_2$ at the point  - $Q$. Let $W_1$ tangent at $P$ intersects $AB$ at point $R$, and the tangent to $W_2$ at $Q$ intersects $CD$ at $S$. $M$ - midpoint $RS$. Prove that $MP=MQ$

2015 Kazakhstan MO grade XI P2
Given convex quadrilateral $ABCD$. $K$ and $M$ are the midpoints of $BC$ and $AD$ respectively. Segments $AK$ and $BM$ intersect at the point $N$, and the segments $KD$ and $CM$ at the point $L$. And quadrilateral $KLMN$ is inscribed. Let the circumscribed circles of triangles $BNK$ and $AMN$ second time intersect at the point $Q$, and circumscribed circles of triangles $KLC$ and $DML$ at the point $P$. Prove that the areas of quadrilaterals of $KLMN$ and $KPMQ$ are equal.

2015 Kazakhstan MO grade XI P4
In triangle $ABC$, point $N$ - base bisector of $C$, point $M$ - the midpoint of side $AB$, and $W$ -  circumcircle of triangle $ABC$. Line $CN$ second time intersects $W$ at point $D$. At segments $AD$ and $BD$, points $K$ and $L$, respectively, so that $\angle ACK = \angle BCL$. Let circumscribed circles of triangles $ACK$ and $BCL$ second time intersect at point $P$, and $Q$ - point of intersection of $DM$ and $KL$. Prove that the points $M, N, P, Q$ lie on a circle.

2016 Kazakhstan MO grade IX P3, grade X P3
Around the triangle $ABC$ a circle is $\omega$ circumscribed ,   $I$  is the intersection point of the bisectors of this triangle. Line $CI$ intersects $\omega$  for the second time at point $P$. Let the circle with diameter $IP$ intersect $AI$, $BI$ and $\omega$  for the second time at points $M$, $N$ and $K$  respectively. Lines  $KN$ and  $AB$ intersect at a point $B_1$ , and the segments $KM$ and  $AB$ intersect  at the point $A_1$. Prove that $\angle ACB = \angle A_1IB_1$.

2016 Kazakhstan MO grade IX P4 
In the triangle $ABC$   from the largest angle $C$, $CH$ let be the  altitude. Lines $HM$ and $HN$, are altitudes of triangles  $ACH$ and  $BCH$ respectively, $HP$ и $HQ$  are  bisectors of triangles  $AMH$ and $BNH$. Let $R$  be  the foot of the perpendicular from the point $H$ on the line $PQ$ . Prove that $R$ is the point of intersection of the bisectors of the triangle $MNH$ .

Incircle of a triangle $ABC$ touches the sides $BC$ and  $AC$ at points  $A_1$ and $B_1$, and the excircle corresponding to the side $AB$ , touching the extensions of these sides in points   $A_2$ and $B_2$  respectively. Suppose that incircle $\triangle ABC$ touches the side $AB$ at the point $K$ . Denote by $O_a$ and $O_b$ the centers of the cicrumcircles of the triangles $A_1A_2K$ and $B_1B_2K$ respectively . Prove that the line  $O_a O_b$ passes through the midpoint of the segment $AB$.

2016 Kazakhstan MO grade XI P3
Circles $\omega_1 , \omega_2$ intersect at points $X,Y$ and they are internally tangent to circle $\Omega$ at points $A,B$,respectively.$AB$ intersect with $\omega_1 , \omega_2$ at points $A_1,B_1$ ,respectively.Another circle is internally tangent to $\omega_1 , \omega_2$ and $A_1B_1$ at $Z$.Prove that $\angle AXZ =\angle BXZ$.

2016 Kazakhstan MO grade XI P4
In isosceles triangle $ABC$($CA=CB$),$CH$ is altitude and $M$ is midpoint of $BH$.Let $K$ be the foot of the perpendicular from $H$ to $AC$ and $L=BK \cap CM$ .Let the perpendicular drawn from $B$ to $BC$ intersects with $HL$ at $N$.Prove that $\angle ACB=2 \angle BCN$.

2017 Kazakhstan MO grade IX P3
On the sides of the triangle $ABC$, rectangles of equal areas $ABLK$, $BCNM$ and $CAQP$ are built externally (points $L, K, N, M, Q, P$ lie outside of the triangle $ABC$, $A(ABLK)=A(BCNM)=A(CAQP)$). Let $X$, $Y$ and $Z$ be the midpoints of the segments $KQ$, $LM$ and $NP$, respectively. Prove that the lines $AX$, $BY$ and $CZ$ intersect at one point.

2017 Kazakhstan MO grade IX P4
The non-isosceles triangle $ABC$ is inscribed in the circle $\omega$ with center $O$. The extension of the bisector $CN$ intersects $\omega$ at the point $M$. Let $MK$ be the height of the triangle $BCM$, $P$ be the middle of the segment $CM$, and $Q$ be the intersection point of the lines $OP$ and $AB$. Suppose that the line $MQ$ crosses $\omega$ for the second time at the point $R$, and $T$ is the intersection point of the lines $BR$ and $MK$. Prove that $NT {\parallel} PK$.

2017 Kazakhstan MO grade X P3
$ABC$ is isosceles triangle. The points $K$ and $N$ lie on the side $AC$, and the points $M$ and $L$ on the side $BC$ so that $AN = CK = CL = BM$. Let the segments $KL$ and $MN$ intersect at $P$. Prove that $\angle RPN = \angle QPK$, where $R$ is the midpoint of $AB$, and $Q$ is the midpoint of the arc $ACB$ of the circumcircle of the triangle $ABC$.

2017 Kazakhstan MO grade X P4. grade XI P4
The acute triangle $ABC$ $(AC> BC)$ is inscribed in a circle with the center at the point $O$, and $CD$ is the diameter of this circle. The point $K$ is on the continuation of the ray $DA$ beyond the point $A$. And the point $L$ is on the segment $BD$ $(DL> LB)$ so that $\angle OKD = \angle BAC$, $\angle OLD = \angle ABC$. Prove that the line $KL$ passes through the midpoint of the segment $AB$.

2017 Kazakhstan MO grade XI P1
The non-isosceles triangle $ABC$ is inscribed in the circle $\omega$. The tangent line to this circle at the point $C$ intersects the line $AB$ at the point $D$. Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$, respectively. The point $M$ is on the side $AB$ such that $\frac{AK}{BL} = \frac{AM}{BM}$. Let the perpendiculars from the point $M$ to the straight lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$ respectively. Prove that  $2\angle CQP=\angle ACB$

2018 Kazakhstan MO grade IX P1
Given a parallelogram $ABCD$. A certain circle passes through points $A$ and $B$ and intersects the segments  $BD$ and $AC$ the second time in the points  $X$ and  $Y$, and the circumscribed circle of the triangle $ADX$ crosses the segment $AC$  for the second time at the point $Z$ . Prove that the segments $AY$ and  $CZ$ are equal.

2018 Kazakhstan MO grade IX P6
On the side  $CD$ of a trapezoid  $ABCD$ there is a point  $M$  such that  $BM=BC$. . Suppose that lines $BM$ and  $AC$ intersect at a point $K$, while lines $DK$ and $BC$  at the point $L$. Prove that the angles $BML$ and $DAM$ are equal.

2018 Kazakhstan MO grade X P1
The trapezium diagonals $ ABCD $ ($ AD \parallel BC $) intersect at the point $ K $. The points $ L $ and $ M $ are marked on the line $ AD $ such that $ A $ lies on the segment $ LD $, $ D $ lies on the segment $ AM $, $ AL = AK $ and $ DM = DK $. Prove that the lines $ CL $ and $ BM $ intersect on the bisector of the angle $ BKC $.

2018 Kazakhstan MO grade X P6
The diagonals of the inscribed convex quadrilateral $ ABCD $ intersect at the point $ O $. Let $ \ell $ be a line dividing the angle $ AOB $ in half. Denote by $ (\ell_1, \ell_2, \ell_3) $ the nondegenerate triangle formed by the lines $ \ell_1, \ell_2, \ell_3 $. Let $ \Delta_1 = (\ell, AB, CD) $ and $ \Delta_2 = (\ell, AD, BC) $. Prove that the circumscribed circles of the triangles $ \Delta_1 $ and $ \Delta_2 $ are tangent to each other.

2018 Kazakhstan MO grade XI P1
In an equilateral trapezoid, the point $O$ is the midpoint of the base $AD$. A circle with a center at a point $O$ and a radius $BO$ is tangent to a straight line $AB$. Let the segment $AC$ intersect this circle at point $K(K \ne C)$, and let $M$ is a point such that $ABCM$ is a parallelogram. The circumscribed circle of a triangle $CMD$ intersects the segment $AC$ at a point $L(L\ne C)$. Prove that $AK=CL$.

2018 Kazakhstan MO grade XI P6
Inside of convex quadrilateral $ABCD$ found a point $M$ such that $\angle AMB=\angle ADM+\angle BCM$ and $\angle AMD=\angle ABM+\angle DCM$. Prove that $$AM\cdot CM+BM\cdot DM\ge \sqrt{AB\cdot BC\cdot CD\cdot DA}.$$

2019 Kazakhstan MO grade IX P2
Given an inscribed convex pentagon $ ABCDE $. The circle centered at $ E $ and radius $ AE $ intersects the segments $ AC $ and $ AD $ in $ X $ and $ Y $, respectively, and the circle centered at $ C $ with radius $ BC $ intersects the segments $ BE $ and $ BD $ at $ Z $ and $ T $, respectively. The lines $ XY $ and $ ZT $ intersect at $ F $. Prove that $ DF $ and $ EC $ are perpendicular.

2019 Kazakhstan MO grade IX P5
In the right-angled triangle $ ABC $, the point $ D $ is symmetric to the point $ C $ with respect to the hypotenuse $ AB $. Let $ M $ be an arbitrary point of the segment $ AC $, and $ P $ be the base of the perpendicular from point $ C $ to the line $ BM $. Point $ H $ is the midpoint of the segment $ CD $. On the segment $ CH $ (inside the angle $ HPB $) there is a point $ N $ such that $ \angle DPH = \angle NPB $. Prove that the points $ M $, $ P $, $ N $ and $ D $ lie on the same circle.

2019 Kazakhstan MO grade X P5
In the circle $ \omega $, the diameter is $ AB $ and the chord $ CD $ is perpendicular. Let $ M $ be any point of the segment $ AC $. Point $ P $ is the base of the perpendicular from point $ C $ to line $ BM $. Let the circle $ \omega_1 $ circumscribed around the triangle $ MPD $ intersect the circumscribed circle of triangle $ CPB $ for the second time at the point $ Q $ (the points $ P $ and $ Q $ lie on opposite sides of the line $ AB $). The line $ CD $ intersects again $ \omega_1 $ at the point $ N $. Prove that $ \angle CQN = \angle BPN $.

The tangent line to the circumcircle of the acute triangle $ ABC $ intersects the lines $ AB $, $ BC $ and $ CA $ at the points $ C '$, $ A' $ and $ B '$, respectively. Let $ H $ be the orthocenter of a triangle $ ABC $. On lines $ A'H $, $ B'H $ and $ C'H $, respectively, points $ A_1 $, $ B_1 $ and $ C_1 $ (other than $ H $) are marked such that $ AH = AA_1 $, $ BH = BB_1 $ and $ CH = CC_1 $. Prove that the circles circumscribed around the triangles $ ABC $ and $ A_1B_1C_1 $ are tangent.

source: matol.kz/nodes/13

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