geometry problems from Kazakhstan Mathematical Olympiads

with aops links in the names

1999 - 2019

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. $

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 $.

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 $.

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 $.

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 $.

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 $.

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 $.

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.

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.

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

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$

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 $KX$

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.

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

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 $\frac{1}{S}\le \frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{2}}$

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 $BN + CM <BC$

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 $3CX=CC_1$

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 $\angle ADC.$

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 $\angle ABP = \angle QBC.$

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 $AB + BF = CF$.

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 $MP=MQ$

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 $AB$.

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 $NT {\parallel} PK$.

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.

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.

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.

source: matol.kz/nodes/13

with aops links in the names

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 $.

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.

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.

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.

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.

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.

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.

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 P1Given 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 $.

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 $.Given a triangle $ ABC $ and a point $ M $ inside it. Prove that $

\min \{MA, MB, MC\} + MA + MB + MC <AB + BC + AC. $

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.

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.

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 P7In 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 $.

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.

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 $.

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 $.

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 , X 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 $.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.

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 $.

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.

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 P6Let 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.

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 $..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 $).

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.

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.

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 $\angle MTB$.

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 P6grade 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 $.

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) $.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.

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.

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 P5The 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. $

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 P5Let $ABC$ 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 $ABC$.

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 P4The 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.

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 P5Suppose 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$.

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

2009 Kazakhstan MO grade IX P5
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.

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

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 $B_2$.

2010 Kazakhstan MO grade IX P1Proved 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

.

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.

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 P5Triangle $ABC$ 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 $XT+YT=XZ+YZ$)

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 P5Let $ABCD$ 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 $XYZ$

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 P2Inscribed 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 $AC = 3 (BC-AB)$.

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 P5Given 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 $O_{1}O_{2}$

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 P4Let $ABCD$ 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 $

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 P5Line $PQ$ 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

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 $

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 $OI$.

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 P5Let $ABCD$ 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 P1Given 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.

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 $BC_2*B_2C = B_2M*C_2N$

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 P5The 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.

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 P4The 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.

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 $AP_1$=$CP_2$

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 $ABC$.

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 P5Circle $W$, 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.

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 P4Given 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.

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 $\angle ACB = \angle A_1IB_1$.

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 P4Circles $\omega_1 , \omega_2$ 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 $\angle AXZ =\angle BXZ$.

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 $\angle ACB=2 \angle BCN$.

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.

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$ABC$ 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.

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 P6The 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 $.

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 P6In 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 $AK=CL$.

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.

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 $.

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

## Δεν υπάρχουν σχόλια:

## Δημοσίευση σχολίου