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Almaty City MO - geometry

3

Almaty City Math Olympiad (Kazakhstan)

2008.2

The diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$ intersect at the point $E$ , $M$ the midpoint of the segment $AE$ and $N$ the midpoint of the segment $CD$ . It is known that the diagonal $BD$ is the bisector of the angle $ABC$ . Prove that quadrilateral $ABCD$ is cyclic if and only if quadrilateral $MBCN$ is cyclic.

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2009.2

The extensions of the sides $AB$ and $CD$ of the inscribed quadrilateral $ABCD$ intersect at the point $P$ , and the extensions of the sides $BC$ and $AD$ at the point $Q$ . Prove that the intersection points of the bisectors of the angles $AQB$ and $BPC$ with the sides of the quadrilateral are vertices of a rhombus.

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2009.4

Given a triangle $ABC$ , in which $AB \ne AC$ . Denote on its sides $AB$ and $AC$ , the points $M$ and $N$ , respectively, such that $BM = CN$ The circumcircle of the triangle $AMN$ intersects the circumcircle of the triangle $ABC$ at $D$ , other than $A$ . Prove that $DM = DN$ .

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2010.3

The circle $\omega$ is circumscribed around the quadrilateral $ABCD$ . The lines $AB$ and $CD$ intersect at the point $K$ , and the lines $AD$ and $BC$ intersect at the point $L$ . The line passing through the center of the circle $\omega$ and perpendicular on $KL$ intersects the lines $KL$ , $CD$ and $AD$ at the points $P$ , $Q$ and $R$ , respectively. Prove that the lines $QL$ , $BP$ and $KR$ intersect at one point.

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2010.4

In $ABC$ , angle bisector $BK$ is drawn. The tangent at $K$ to the circumcircle $\omega$ of the triangle $ABK$ intersects the side $BC$ at the point $L$ . The line $AL$ intersects $\omega$ at $M$ . Prove that the line $BM$ passes through the midpoint of the segment $KL$ .

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2011.4

The tangents $SA$ and $SB$ are drawn to the circle $\omega$ with center $O$ , from the point $S$ . Points $C$ and $C '$ on the circle $\omega$ such that $AC \parallel OB$ and $CC'$ is the diameter of $\omega$ . Let lines $BC$ and $SA$ intersect at $K$ , and lines $KC '$ and $AC$ at $M$ . Prove that in the triangle $MKC$ the altitude from the vertex $M$ divides the altitude from the vertex $C$ in half if the angle $BMK$ is right.

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2012.1

On the coordinate plane $xOy$ , a parabola $y = {{x} ^ {2}}$ is drawn. Let $A$ , $B$ and $C$ be different points of this parabola. We define the point ${{A} _ {1}}$ as the intersection point of the line $BC$ and the $Oy$ axis. Similarly, we define the points ${{B} _ {1}}$ and ${{C} _ {1}}$ . Prove that the sum of the distance from $A$ , $B$ and $C$ to the $Ox$ axis is greater than the sum of the distance from ${{A} _ {1}}$ , ${{B} _ {1}}$ and ${{C} _ {1}}$ to the $Ox$ axis.

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2012.3

In the isosceles triangle $ABC$ $(BC = AC)$ on the bisector of $BN$ there was a point $K$ such that $BK = KC$ and $KN = NA$ . Find the angles of the triangle $ABC$ .

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2013.3

On a line containing the altitude $A {{A} _ {1}}$ of the triangle $ABC$ $(\angle B \ne 90^\circ)$ , a point $F$ is taken that is different from $A$ and ${{A} _ {1}}$ such that the lines $BF$ and $CF$ intersect the lines $AC$ and $AB$ at the points ${{B} _ {1}}$ and ${{C} _ {1}}$ respectively. Perpendiculars $BP$ , $BQ$ , $FS$ , $FR$ on lines ${{A} _ {1}} {{B} _ {1}}$ and ${{A}_{1}}{{C}_{1}}$ are pass through points $B$ and $F$ . Prove that the lines $PQ$ , $SR$ and $B {{B} _ {1}}$ intersect at the same point.
a) Solve the problem when $F$ is the intersection point of the altitudes of the triangle $ABC$ .
b) Solve the problem for an arbitrary point $F$ .

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2014.1

The line $l$ is the tangent to the circle circumscribed around the acute-angled triangle $ABC$ , drawn at the point $B$ . The point $K$ is the projection of the orthocenter of the triangle onto the line $l$ , and the point $L$ is the midpoint of the side $AC$ . Prove that the triangle $BKL$ is isosceles.

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2015.2

The altitudes $AA_1$ and $CC_1$ of the acute-angled triangle $ABC$ intersect at the point $H$ . On the altitude $AA_1$ , lies point $P$ such that $A_1P = AH$ . On the altitude $CC_1$ , lies point $Q$ such that $C_1Q = CH$ . Prove that the perpendiculars on the lines $AA_1$ and $CC_1$ passing through the points $P$ and $Q$ , respectively, intersect on the circumcircle of the triangle $ABC$

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2016.3

The altitudes of $A {{A} _ {1}}$ , $B {{B} _ {1}}$ and $C {{C} _ {1}}$ are drawn in the acute triangle $ABC$ . To the circumcircle of the triangle $ABC$ , the tangents at the points $A$ and $C$ intersecting at the point $Q$ are drawn. A straight line passing through the midpoint of the side $AC$ and the orthocenter of the triangle $ABC$ intersects the line ${{A}_{1}} {{C} _ {1}}$ at the point $F$ . Prove that the points $Q$ , ${{B} _ {1}}$ , and $F$ lie on the same line.

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2017.3

The circle $\omega$ passing through the vertices $A$ and $B$ of the triangle $ABC$ intersects the sides $AC$ and $BC$ at the points $E$ and $F$ , respectively. The circle $\Gamma$ tangent to the segment $EF$ at the point $P$ and the arc $AB$ of the circumcircle of the triangle $ABC$ at the point $Q$ . Prove that $C$ , $P$ , $Q$ lie on one line.

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2018.2

In the triangle $ABC$ : $BC\ge CA \ge AB$ . From the vertices $B$ and $C$ let $BK$ and $CL$ be angle bisectors respectively. Inside the triangle $AKL$ a point $X$ is selected from which perpendiculars $XY$ and $XZ$ are drawn on $AB$ and $AC$ respectively. Prove that $XY+YZ+ZX < AC$ .

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2019.4

The point $M$ is marked on the segment $BC$ of the triangle $ABC$ . Let $I$ , $K$ , $L$ be the centers of the inscribed circles $\omega_1$ , $\omega_2$ , $\omega_3$ of the triangles $ABC$ , $ABM$ , $ACM$ , respectively. A common external tangent to the circles $\omega_2$ and $\omega_3$ , different from the line $BC$ , intersects the segment $AM$ at the point $J$ . It is known that the points $I$ and $J$ do not coincide and lie inside $\triangle AKL$ . Prove that in the triangle $AKL$ the points $I$ and $J$ are isogonally conjugate. (The interior points $P$ and $P '$ of the triangle $ABC$ are called isogonally conjugate if $\angle ABP = \angle CBP '$ , $\angle BAP = \angle CAP '$ , $\angle BCP = \angle ACP '$ .)

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2021.4

The quadrilateral $ABCD$ is inscribed in the circle $\Gamma$ . The diagonals $AC$ and $BD$ meet at the point $E$ . Let $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEB$ and $CED$ , respectively. On the arc $AB$ , not containing the point $E$ , of the circle $\omega_1$ , the point $P$ is selected, and on the arc $CD$ , not containing the point $E$ , of the circle $\omega_2$ , the point $Q$ is selected so, that $\angle AEP = \angle QED$ . The segment $PQ$ intersects $\Gamma$ at the points $X$ and $Y$ . Prove that $PX = QY$ .

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2023.1

On sides $BC,$ $CA,$ $AB$ of triangle $ABC$ points $K,L,M$ are chosen respectively, and inside the triangle point $P$ is chosen so that $PL \parallel BC,$ $PM \parallel CA,$ $PK \parallel AB$ . Could it turn out that all three trapezoids $AMPL,$ $BKPM$ , $CLPK$ are tangential?

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2024.4

Inside the convex quadrilateral $ABCD$ , let $G$ be the point of intersection of the medians of triangle $ABC$ . It turned out that $G \in [BD]$ and $\angle DCG = \angle BAC$ . From point $D$ a perpendicular $DE$ is dropped on the segment $CG$ . Prove that $\frac{CG}{DE} \ge \frac{2\sqrt 3}{3}.$

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