## Σελίδες

### Kazakhstan 1999 -2020 IX-XI 115p

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

1999 - 2020

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 circumscribed 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 circumscribed 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
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 $ABC$.

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.

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.

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 circumscribes 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.

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 $w$ — circumscribed 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 circumscribed —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 $W$, circumscribed around 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.

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 $ABC$.

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.

2020 Kazakhstan MO grade IX P4
The incircle of the triangle $ABC$ touches the sides of $AB, BC, CA$ at points $C_0, A_0, B_0$, respectively. Let the point $M$ be the midpoint of the segment connecting the vertex $C_0$ with the point of intersection of the altitudes of the triangle $A_0B_0C_0$, point $N$ be the midpoint of the arc $ACB$ of the circumscribed circle of the triangle $ABC$. Prove that line $MN$ passes through the center of incircle of triangle $ABC$.

A point $N$ is marked on the median $CM$ of the triangle $ABC$ so that $MN \cdot MC = AB ^ 2/4$. Lines $AN$ and $BN$ intersect the circumcircle $\triangle ABC$ for the second time at points $P$ and $Q$, respectively. $R$ is the point of segment $PQ$, nearest to $Q$, such that $\angle NRC = \angle BNC$. $S$ is the point of the segment $PQ$ closest to $P$ such that $\angle NSC = \angle ANC$. Prove that $RN = SN$.