Almaty Olympiad 2008-18 (Kazakhstan) 14p

geometry problems from Almaty Olympiads (final stage, from Kazakhstan)
with aops links in the names


2008 - 2018

2008 Almaty Olympiad P2
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.

2009 Almaty Olympiad P2
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.

2009 Almaty Olympiad P4
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 $.

2010 Almaty Olympiad P3
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.

2010 Almaty Olympiad P4
In $ ABC $  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 middle of the segment $ KL $.

2011 Almaty Olympiad P4
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 straight.

2012 Almaty Olympiad P1
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.

2012 Almaty Olympiad P3
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 $.

2013 Almaty Olympiad P3
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 $.

2014 Almaty Olympiad P1
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.

2015 Almaty Olympiad P2
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 $.

2016 Almaty Olympiad P3
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.

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.

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 dropped to $AB$ and  $AC$ respectively. Prove that $XY+YZ+ZX < AC$ .

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