Zhautykov City 2001-19 VII-IX (Kazakhstan) 91p

geometry problems from Zhautykov City Olympiads (from Kazakhstan)         
with aops links in the names


2001- 2019 


Two circles $ {{K} _{1}} $ and $ {{K} _{2}} $ intersect at points $ A $ and $ B $. A common tangent is drawn that touches the circles $ {{K} _{1}} $ and $ {{K} _{2}} $ respectively at the points $ {{C} _{1}} $ and $ {{C} _{2}} $. Prove that the triangles $ AB {{C} _{1}} $ and $ AB {{C} _{2}} $ are equal.

2001 Zhautykov City MO grade IX P2
Given an inscribed quadrangle $ ABCD $, whose sides intersect at points $ K $ and $ M $. The bisectors of the angles $ K $ and $ M $ intersect the sides of $ ABCD $ at four points. Prove that these four points form a rhombus.

2001 Zhautykov City MO grade X P1
In the triangle $ ABC $,  $ \angle A = 3 \angle C $. The point $ D $ on the side of $ BC $ has the property that $ \angle ADC = 2 \angle C $. Prove that $ AB + AD = BC $.

2001 Zhautykov City MO grade XI P2
In the triangle $ ABC $, the point $ D $ is taken in such a way that $ \angle BDC = 2 \angle BAC $. On the segment $ CD $, a point $ E $ is chosen such that $ BD + DE = AE $. Prove that $ \angle AEC = 2 \angle ABC $.

2002 Zhautykov City MO grade VII P4
Prove that the perpendicular raised from the middle of the hypotenuse of a right triangle with an angle of $ 30 {} ^\circ $ is equal to $ \dfrac {1} {3} $ of a larger leg.

2002 Zhautykov City MO grade VIII P1
On the sides of $ AD $ and $ DC $ of the rhombus $ ABCD $, regular triangles $ AKD $ and $ DMC $ are constructed so that the point $ K $ lies on the same side of $ AD $ as the line $ BC $, and the point $ M $ , on the other side of $ DC $ than $ AB $. Prove that the points $ B $, $ K $ and $ M $ lie on the same line

2002 Zhautykov City MO grade IX P3
The tangents to the circle circumscribed around the triangle $ ABC $ drawn at the points $ A $ and $ B $ intersect at the point $ P $. Prove that the line $ PC $ intersects the side $ AB $ at the point $ K $ dividing it with respect to $ A {{C} ^{2}}: B {{C} ^{2}} $.

2003 Zhautykov City MO grade VII P3
In the isosceles triangle $ ABC $ ($ AB = AC $, $ \angle BAC = 30 {}^\circ $) on the side $ AB $ and the median $ AD $, respectively, the points $ Q $ and $ P $ are chosen so that $ PC = PQ $ ($ P \ne Q $). Find $ \angle PQC $.

2003 Zhautykov City MO grade VIII-IX P2
Let $ AM $ and $ BN $ be the heights of an acute-angled triangle $ ABC $ ($ \angle ACB \ne 45 {} ^\circ $). The points $ K $ and $ T $ are marked on the rays $ MA $ and $ NB $, respectively, so that $ MK = MB $ and $ NT = NA $. Prove that $ KT \parallel MN $.

2003 Zhautykov City MO grade X-XI P2
In the rhombus $ ABCD $, the angle at the vertex $ B $ is $ 60 {} ^\circ $. Inside $ \Delta ADC $, the point $ M $ is chosen so that $ \angle AMC = 120 {} ^\circ $. Let $ P $ and $ Q $ be the intersection points of the lines $ BA $ and $ CM $, $ BC $ and $ AM $, respectively. Prove that the point $ D $ lies on the line $ PQ $.

2004 Zhautykov City MO grade VII P4
Think of three triangles from which you can make (without overlapping) a triangle, a convex quadrangle, and a convex pentagon (all three triangles must be used each time, the triangles are allowed to be rotated).

2004 Zhautykov City MO grade VII P5
A square with side $1$ is given. Find the set of all points whose sum of distances from the sides of this square or their extensions is $4$.

On the lateral side of $ BC $ an isosceles triangle $ ABC $ with a vertex at the point $ C $, the point $ M $ is taken, and on the segment $ MC $ - the point $ N $ so that $ MN = AM $. It is known that angles $ BAM $ and $ NAC $ are equal. Find the angle value of $ MAC $.

2004 Zhautykov City MO grade VIII P5
In the right triangle $ ABC $ on the legs $ AB $ and $ BC $ ($ 2BC> AB> BC $), the points $ D $ and $ E $ are taken so that $ AD = CB $, $ BD = CE $. Prove that the angle between the segments $ AE $ and $ CD $ is $ 45 {} ^ \circ $.

A quadrilateral is given. $ A, B, C, D $ are the successive midpoints of its sides. $ P, Q $ are the midpoints of the diagonals. Prove that $ \vartriangle BCP = \vartriangle ADQ $.

2004 Zhautykov City MO grade IX P8
Inside the given $ \vartriangle ABC $, find a point $ O $ such that the areas of the triangles $ AOB $, $ BOC $ and $ COA $ are referred to as $ 1: 2: 3 $.

2004 Zhautykov City MO grade IX P9
In the quadrangle $ ABCD $, the diagonals intersect at the point $ O $ with $ \angle AOD = 120 ^\circ $, $ AO = OD $. Let $ E $ be an arbitrary point on the side of $ BC $. The points $ K $ and $ P $ are taken respectively on the sides of $ AB $ and $ CD $ so that $ KE \parallel AC $ and $ PE \parallel BD $. Prove that the center of the circumscribed circle around $ \vartriangle KEP $ is located on the side of $ AD $.

2005 Zhautykov City MO grade VII P6
In the triangle $ ABC $, the bisector $ AE $ is equal to the segment $ EC $. Find the angle $ ABC $ if $ AC = 2AB $.

2005 Zhautykov City MO grade VIII P6
On the side $ BC $ of the triangle $ ABC $, the point $ K $ is selected. It turned out that the segment $ AK $ intersects the median $ BD $ at the point $ E $ so that $ AE = BC $. Prove that $ BK = KE $.

2005 Zhautykov City MO grade IX P4
Prove that if the sides of the triangle form an arithmetic progression, then the line segment connecting the intersection point of the medians with the center of the inscribed circle is parallel to the middle side.

2005 Zhautykov City MO grade IX P5
In convex quadrangle $ ABCD $, $ \angle A = \angle D $. The perpendicular bisectors of the sides $ AB $ and $ CD $ intersect at a point $ P $ lying on the side $ AD $. Prove that the diagonals $ AC $ and $ BD $ are equal.

2006 Zhautykov City MO grade VII P5
Let $ AF $ be the median of the triangle $ ABC $. $ D $ is the midpoint of the segment $ AF $, $ E $ is the intersection point of the line $ CD $ with the side $ AB $. It turned out that $ BD = BF = CF $. Prove that $ AE = DE $

2006 Zhautykov City MO grade VII P10
Given an isosceles triangle with an angle of $ 20 {} ^\circ $ at the vertex. Prove that its side is more than twice the base.

2006 Zhautykov City MO grade VIII P3
Prove that any parallelogram can be cut into exactly $9$ isosceles triangles.

2006 Zhautykov City MO grade VIII P5
Let the triangle $ M $ have the middle $ AB $ and $ D \in AC $ of the base of the bisector of the angle $ \angle ABC $. Prove that $ AB = 3BC $ if $ MD \bot BD $.

2006 Zhautykov City MO grade VIII P9
The bisectors $ A {{A} _{1}} $ and $ C {{C} _{1}} $ are drawn in the triangle $ ABC $. $ M $ and $ K $ are the bases of perpendiculars dropped from the point $ B $ to the lines $ A {{A} _{1}} $ and $ C {{C} _{1}} $. Prove that $ MK \parallel AC $.

2006 Zhautykov City MO grade IX P2
Prove that any triangle can be cut into $3$ polygons, which make up a right triangle (parts cannot be turned over)

2006 Zhautykov City MO grade IX P3
In the non-isosceles acute-angled triangle $ ABC $, the height $ BD $ is drawn. On the extension of $ DB $ for the point $ B $, the point $ K $ is chosen so that $ \angle KAC = \angle BCA $. Prove that the circle passing through the point $ B $ and tangnent to the line $ AC $ at the point $ C $ intersects $ BD $ in the orthocenter of the triangle $ AKC $

2006 Zhautykov City MO grade IX P10
Given a convex quadrangle $ ABCD $, in which $ \angle A = 90 ^\circ $, and the vertex $ C $ is removed from the lines $ AB $ and $ AD $ by distances equal to the lengths of the segments $ AB $ and $ AD $, respectively . Prove that the diagonals of the quadrangle are mutually perpendicular.

2007 Zhautykov City MO grade VII P4
In the right triangle $ ABC $, let $ K $ be the midpoint of the hypotenuse $ AB $. On the leg $ BC $, the point $ M $ is chosen, so that $ BM = 2MC $. Prove that $ \angle MAB = \angle MKC $.

2007 Zhautykov City MO grade VIII P4
The non parallel sides of a trapezoid have ratio of lengths $1: 2$. The sum of the angles of the larger base is $ 120 {} ^\circ $. Find the angles of of this trapezoid.

2007 Zhautykov City MO grade VIII P8
The diagonals $ AC $ and $ BD $ of the quadrangle $ ABCD $ are equal and intersect at the point $ F $.Prove that the line connecting the midpoints of the sides $ BC $ and $ AD $ is perpendicular to the bisector of the angle $ \angle CFD $.

2007 Zhautykov City MO grade IX P4
In the quadrangle $ ABCD $, the sides of $ AD $ and $ CD $ are equal, $ \angle BCD = 60 {} ^\circ $, $ \angle BAC = 30 {} ^\circ $. Prove that the sides of $ BC $ and $ CD $ are also equal.

2007 Zhautykov City MO grade IX P8
The points $ K $ and $ L $ are marked on the side $ AB $ of the triangle $ ABC $ (the point $ K $ lies between the points $ A $ and $ L $). It is known that $ AK \cdot LB = AB \cdot KL $ and $ \angle LCK = \angle LCB $. Prove that the angle $ ACL $ is a straight line.

2008 Zhautykov City MO grade VII P4
Given rectangle $ ABCD $. Let point $ M $ be the midpoint of the side $ BC $, point $ N $ be the midpoint of the side $ CD $, point $ K $ be the intersection point of the segments $ BN $ and $ MD $ (see. Fig.). Prove that $ \angle MKB = \angle MAN $.

2008 Zhautykov City MO grade VII P8
In the right-angled triangle $ ABC $, the bisectors $ AP $ and $ BQ $ are drawn from the vertices of acute angles. The points $ D $ and $ E $ are the bases of the perpendiculars dropped from $ Q $ and $ P $ to the hypotenuse $ AB $. Find the angle $ DCE $

2008 Zhautykov City MO grade VIII P4
In the isosceles triangle $ ABC $ $ (AC = BC) $, the point $ D $ is marked on the side $ AC $ so that the triangle $ ADK $ is isosceles, where $ K $ is the intersection point of the segment $ BD $ and the height $ AH $. Find the value of the angle $ DBA $.

2008 Zhautykov City MO grade IX P4
On a circle inscribed in an equilateral triangle $ ABC $, the point $ P $ is taken. The segment $ AP $ once again intersects the circle at the point $ Q $ so that $ AQ = QP $. Find the angle of $ BPC $.

2008 Zhautykov City MO grade IX P8
On the side $ BC $ of the triangle $ ABC $, the point $ K $ is marked, so that $ \dfrac {BK} {KC} \le 1 $. The point $ M $ is the midpoint of the side $ AC $, and $ N $ is such a point on the line  $ AC $ such that $ BN \parallel KM $. Prove that the segment $ KN $ divides the triangle $ ABC $ into two equal figures.

2009 Zhautykov City MO grade VII P4
A triangle, one of whose angles is $ 40 {}^\circ $, was cut along its bisectors into six triangles, among which there are right ones. What can be the other corners of the original triangle?

2009 Zhautykov City MO grade VII P8
On the sides $ AB $ and $ BC $ of the square $ ABCD $, as is the basis, isosceles triangles $ ABP $ and $ BCQ $ are constructed with an angle of $ 80 {}^\circ $ at the vertex, with the point $ P $ lying inside the square and the point $ Q $ out of the square. Find the angle between the lines $ PQ $ and $ BC $.

2009 Zhautykov City MO grade VIII P4
The point $ D $ is the midpoint of the side $ AC $ of the triangle $ ABC $. On the $ BC $ side, a point $ E $ is chosen such that $ \angle BEA = \angle CED $. Find the ratio of the lengths of $ AE: DE $.

2009 Zhautykov City MO grade VIII P8
In the trapezoid $ ABCD $ ($ BC \parallel AD $), the bisectors of the angles $ A $ and $ B $ intersect at the point $ M $, and the bisectors of the angles $ C $ and $ D $ intersect at the point $ N $. $ BC = a $, $ AD = b $, $ AB = c $, $ CD = d $. Find the length of the segment $ MN $.

2009 Zhautykov City MO grade IX P4
In a convex quadrangle $ ABCD $ ,  $ \angle BAC = \angle DBC = 30 {} ^\circ, \angle BCA = 20 {} ^\circ $ and $ \angle BDC = 70 {} ^\circ $. Prove that $ ABCD $ is a trapezoid.

2010 Zhautykov City MO grade VII P4
The point $ M $ is the midpoint of the side $ AC $ of the triangle $ ABC $. The point $ D $ on the side of $ BC $ is such that $ \angle BMA = \angle DMC $. It turned out that $ CD + DM = BM $. Prove that $ \angle ACB + \angle ABM = \angle BAC $.

2010 Zhautykov City MO grade VIII P4
On the sides $ AB $, $ BC $, $ CD $ and $ DA $ of the square $ ABCD $, the points $ {{A} _{1}}, {{B} _{1}}, {{C} _{ 1}} $ and $ {{D} _ {1}} $ respectively. Prove that if the segments $ {{A} _{1}} {{C} _{1}} $ and $ {{B} _{1}} {{D} _{1}} $ are perpendicular, then $ A {{A} _{1}} + C {{C} _{1}} = B {{B} _{1}} + D {{D} _{1}} $.

2010 Zhautykov City MO grade VIII P10
In the triangle $ ABC $, the median drawn from the vertex $ A $ to the side $ BC $ is four times smaller than the side $ AB $ and forms an angle of $ 60 {} ^\circ $ with it. Find the largest angle of the given triangle.

2010 Zhautykov City MO grade IX P3
On the sides $ BC $ and $ AB $ of an acute-angled triangle $ ABC $, the points $ {{A} _{1}} $ and $ {{C} _{1}} $ are selected. The segments $ A {{A} _{1}} $ and $ C {{C} _{1}} $ intersect at the point $ K $. The circumscribed circles of the triangles $ A {{A} _{1}} B $ and $ C {{C} _{1}} B $ intersect at the point $ P $. It turned out that the point $ P $ is the center of the inscribed circle of the triangle $ AKC $. Prove that $ P $ is the orthocenter of the triangle $ ABC $

2011 Zhautykov City MO grade VII P7
$ O $ is the center of the equilateral triangle $ ABC $. Find the set of points $ X $ such that any line passing through $ X $ intersects the segment $ AB $ or $ OC $.

2011 Zhautykov City MO grade VIII P4
In an acute-angled triangle $ ABC $, the angle $ A $ is $ 60 {} ^\circ $. Prove that the bisector of one of the angles formed by the heights $ BK $ and $ CL $ passes through the center of the circumscribed circle.

2011 Zhautykov City MO grade VIII P8
A rectangle is inscribed in a rectangle (with a vertex on each side). Prove that its perimeter is not less than twice the diagonal of the rectangle.

2011 Zhautykov City MO grade IX P2
The polygon circumscribed around a circle of radius $ r $ is somehow cut into triangles. Prove that the sum of the radii of the inscribed circles of these triangles is greater than $ r $.

2011 Zhautykov City MO grade IX P6
We draw from apoint $ P $ of the bisector of the angle $ A $ of the triangle $ ABC $ the perpendiculars $ P {{A} _{1}} $, $ P {{B} _{1}} $, $ P {{C} _{1}} $ on his sides $ BC $, $ CA $ and $ AB $ respectively. Let $ R $ be the intersection point of the lines $ P {{A} _{1}} $ and $ {{B} _{1}} {{C} _{1}} $. Prove that the line $ AR $ divides the side $ BC $ in half.


2012 Zhautykov City MO grade VII P8
In the acute triangle $ ABC $, the points $ P $, $ Q $ and $ R $ are marked on the sides $ AB $, $ BC $ and $ CA $, respectively, so $ BP = PQ = QR = RC $. We cut out the triangles $ BPQ $, $ PQR $, $ QRC $ and arrange them sequentially so that the bases lie on one straight line, with the second triangle turning upside down so that its vertex $ Q $ also looks up. Prove that the vertices of these three isosceles triangles lie on the same line.

2012 Zhautykov City MO grade VIII P4
In an acute-angled triangle $ ABC $, the point $ M $ is the middle of the side $ BC $ and the points $ N $ and $ H $ are the bases of the heights drawn to the sides $ AB $ and $ AC $, respectively. It is known that $ \angle NMH = \angle ABC $ and $ AC = 8 $ cm. Find the length of the segment $ NH $.

2012 Zhautykov City MO grade VIII P8
On the side of $ AD $ parallelogram $ ABCD $, the point $ R $ is taken, and on the sides $ AB $ and $ CD $ the point $ P $ and $ Q $ respectively, so that the segments $ PR $ and $ QR $ are parallel to the diagonals of the parallelogram. Prove that the areas of the triangles $ PBR $ and $ QCR $ are equal.

2012 Zhautykov City MO grade IX P2
The pentagon $ ABCDE $ is inscribed in a circle. It is known that the distances from the point $ E $ to the lines $ AB $, $ BC $, $ CD $ are different divisors of the number $2012$. It could seem that the distance from the point $ E $ to the line $ AD $ is also a divisor of the number $2012$ ?

2012 Zhautykov City MO grade IX P7
Two points $ A $ and $ B $ are given on the plane. Let $ C $ be some point equidistant from $ A $ and $ B $. We construct a sequence of points $ {{C} _{1}} = C, {{C} _{2}}, {{C} _{3}}, \ldots, {{C} _{n}}, { {C} _{n + 1}}, \ldots $, where $ {{C} _{n + 1}} $ is the center of the circle circumscribed around the triangle $ A {{C} _{n}} B $. At what position of the point $ C $:
a) the point $ {{C} _{n}} $ will fall in the middle of the segment $ AB $ (while $ {{C} _{n + 1}} $ and further members of the sequence are not defined),
 b) does the point $ {{C} _{n}} $ coincide with $ C $?

2013 Zhautykov City MO grade VII P2
Given a triangle is $ ABC $, in which $ \angle ABC = 70 {} ^\circ $, $ \angle ACB = 50 {} ^\circ $. Points $ M $ and $ N $ are marked on the sides of $ AB $ and $ AC $ such that $ \angle MCB = 40 {} ^\circ $ and $ \angle NBC = 50 {} ^\circ $. Find the angle $ \angle NMC $.

2013 Zhautykov City MO grade VII P6
On the base $ BC $ triangle of $ ABC $, find a point $ X $ such that the circles inscribed in the triangles $ ABX $ and $ ACX $ have a common point.

2013 Zhautykov City MO grade VIII P4
In the triangle $ ABC $, by $ A {{A} _{1}} $, $ B {{B} _{1}} $ and $ C {{C} _{1}} $ we denote the heights, and by $ A {{A} _{2}} $, $ B {{B} _{2}} $ and $ C {{C} _{2}} $ are medians. Prove that the length of the broken line $ {{A} _{2}} {{B} _{1}} {{C} _{2}} {{A} _{1}} {{B} _{2} } {{C} _{1}} {{A} _{2}} $ is equal to the perimeter of the triangle $ ABC $.

2013 Zhautykov City MO grade VIII P8
The following relations hold in the convex quadrangle $ ABCD $: $ \angle DAB = \angle ABC = 60 {} ^\circ $ and $ \angle CAB = \angle CBD $. Prove that $ AD + CB = AB $.

2013 Zhautykov City MO grade IX P6
Points are marked on the sides of the acute triangle $ ABC $ so that on the side $ BC $ there are points $ {{A} _{1}} $ between $ {{A} _ {2}} $ and $ C $: $ 6B {{A} _{2}} = 3A_2A_1 = 2A_1C $, on the $ CA $ side there are the points $ {{B} _{1}} $ between $ {{B} _{2}} $ and $ C $: $ C {{B} _{1}} = 2 {{B} _{1}} {{B} _{2}} = {{B} _{2}} A $, on the side of $ AB $ - points $ {{C} _{1}} $ between $ {{C} _{2}} $ and $ A $: $ 14A {{C} _{1}} = 6 {{C} _{1} } {{C} _{2}} = 21 {{C} _{2}} B $. Let $ M, N, K $ be the orthocenters of triangles $ {{C} _{2}} B {{A} _{2}} $, $ {{A} _{1}} C {{B} _{ 1}} $, $ {{B} _{2}} A {{C} _{1}} $. Find the polygon area $ {{C} _{2}} M {{A} _{2}} {{A} _{1}} N {{B} _{1}} {{B} _{2} } K {{C} _{1}} $ if $ \angle CAB = 60 {} ^\circ $, $ \angle ABC = 45 {} ^\circ $ and the area of the triangle $ ABC $ is $144$.

2014 Zhautykov City MO grade VII P7
Let a non-equilateral triangle $ ABC $ be given. The point $ G $ and $ I $ is the intersection point of the medians and bisectors of the triangle $ ABC $, respectively. Prove that at least one of the following three inequalities $ AI> AG $, $ BI> BG $, $ CI> CG $ always holds.

2014 Zhautykov City MO grade VIII P4
On the legs $ AC $ and $ BC $ of an isosceles right triangle $ ABC $, the points $ D $ and $ E $ are marked, respectively, so that $ CD = CE $. Perpendiculars to the line $ AE $ passing through the points $ C $ and $ D $ intersect the side $ AB $ at the points $ P $ and $ Q $. Prove that $ BP = PQ $.

2014 Zhautykov City MO grade VIII P8
Let $ AD $ be the median of the triangle $ ABC $, with $ \angle ADB = 45 {}^\circ $ and $ \angle ACB = 30{}^\circ $. Find the value of the angle $ BAD $.

2014 Zhautykov City MO grade IX P3
In the convex quadrilateral $ ABCD $ on the diagonal $ AC $ a point $ M $ is marked. Through point $ M $ we draw lines $ {{l}_{1}} $ and $ {{l}_{2}} $ such that $ {{l} _ {1}} \parallel AB $ and $ { {l} _ {2}} \parallel CD $. We set $ P $ as the intersection point of $ {{l} _ {1}} $ and $ CB $, $ Q $ is the intersection point of the line $ {{l} _ {2}} $ and $ AD $. Prove that the middle of the segment $ PQ $ lies on $ FE $, where $ F $ is the middle of $ DC $, $ E $ is the middle of $ AB $.

2014 Zhautykov City MO grade IX P8
Let different points $ {{A} _ {1}}, {{A} _ {2}}, \ldots, {{A} _ {2014}} $ be given, no three of which lie on the same line. Let there exist points $ P $ and $ Q $ such that $ {{A} _ {1}} P + {{A} _ {2}} P + \ldots + {{A} _ {2014}} P = { {A} _ {1}} Q + {{A} _ {2}} Q + \ldots + {{A} _ {2014}} Q = 2013. $ Prove that there exists a point $ K $ such that $ {{A} _ {1}} K + {{A} _ {2}} K + \ldots + {{A} _ {2014}} K <2013. $

2015 Zhautykov City MO grade VII P4
Let a triangle $ ABC $ be given. On the $ BC $ side, the point $ A_1 $ is selected, on the $ BA $ side, the $ C_1 $ point is selected. Let $ P $, $ Q $, $ D $ be the midpoints of $ A_1C $, $ C_1A $, $ AC $, respectively. On the ray $ DP $, the point $ E $ is chosen in such a way that $ DE = 2DP $, on the ray $ DQ $ the point $ F $ is chosen so that $ DF = 2DQ $. Prove that $ FA_1 = EC_1 $

2015 Zhautykov City MO grade VIII P4
The heights $ AA_1 $ and $ BB_1 $ of the triangle $ ABC $ intersect at the point $ H $. The points $ X $ and $ Y $ are the midpoints of the segments $ AB $ and $ CH $, respectively. Prove that $ XY $ and $ A_1B_1 $ are perpendicular.

2015 Zhautykov City MO grade IX P2
Let a triangle $ ABC $, $ BC <AB $ be given. Let $ E $, $ D $ be the midpoints of $ BA $, $ AC $, respectively. On the ray $ DE $, the point $ F $ is chosen so that $ DF = 2DE $. Prove that $ 2FA_1 <AB + BC + CA $, where $ A_1 $ is an arbitrary point of the segment $ BC $.

2015 Zhautykov City MO grade IX P8
Let a triangle $ ABC $ be given. On the sides $ AB $, $ BC $, $ CA $, the points $ C_1 $, $ A_1 $, $ B_1 $ are marked, respectively, such that $AA_1,BB_1,CC_1$ are concurrent. Let $ E $ be the base of the height dropped from the point $ A_1 $ to the line $ B_1C_1 $. Prove that $ EA_1 $ is the bisector of $ BEC $. (added the red letters, in order the problem to be correct)

2016 Zhautykov City MO grade VII P4
In the triangle $ ABC $ on the side of $ AC $, the points $ D $ and $ E $ are marked, such that $ AD = DE = EC $. Could it be that $ \angle ABD = \angle DBE = \angle EBC $?

2016 Zhautykov City MO grade VIII P5
In the triangle $ ABC $, heights of $ AD $ and $ BE $ are drawn. The bisector of the angle $ BEC $ intersects the line $ AD $ at the point $ M $, and the bisector of the angle $ ADC $ intersects $ BE $ at the point $ N $. Prove that $ MN \parallel AB $.

2016 Zhautykov City MO grade IX P4
In the acute-angled non-isosceles triangle $ ABC $, the point $ H $ is its orthocenter, $ M $ is the middle of $ AB $, $ N $ is the middle of $ CH $. Let the lines $ AN $ and $ CM $ intersect at $ L $. Prove that $ \angle L {{A} _ {1}} C = \angle ABH $, where $ {{A} _ {1}} $ is the base of the height from the vertex $ A $ of the triangle $ ABC $.

The triangle is $ ABC $. The point $ K $ is taken on the side $ AB $, and the point $ L $ is taken on the side $ AC $ so that $\angle ACB + \angle AKL = 50 {}^\circ $ and $ \angle ABC + \angle ALK = 70 {}^\circ $. What could be the angle of $ BAC $ ?

In an isosceles right-angled triangle $ ABC $ on the legs $ AC $ and $ BC $, the points $ K $ and $ L $ are taken, respectively, so that $ AK / KC = 4/1 $ and $ CL / BL = 3/2 $. Let $ KML $ also be an isosceles right triangle, and $ O $ be the midpoint of its hypotenuse $ MK $. Prove that the point $ O $ lies on the external or internal bisector of the angle $ ACB $.

2017 Zhautykov City MO grade VIII P2
Tangents $ SA $ and $ SB $ are drawn to a circle with center at point $ O $ from point $ S $. On the circle, a point $ C $ is chosen, which is different from the point $ A $, so that the lines $ AC $ and $ SO $ are parallel. Prove that the point $ O $ lies on the line $ BC $.

2017 Zhautykov City MO grade VIII P7
$ R $ and $ r $ are given the radii of the circumscribed and inscribed circles of the triangle $ ABC $, and $ I $ is the center of the inscribed circle. Define the point $ {{A} _ {1}} $ as a point symmetric to the point $ I $ with respect to the perpendicular bisector of the segment $ BC $. Similarly, we define the points $ {{B} _ {1}} $ and $ {{C} _ {1}} $. Prove that the triangles $ ABC $ and $ {{A} _ {1}} {{B} _ {1}} {{C} _ {1}} $ are similar, and find the similarity ratio.

2017 Zhautykov City MO grade IX P3
A tangent $ AB $ is drawn to a circle with a center at the point $ O $ from the point $ A $. Point $ C $ lies on a circle, different from point $ B $ and $ AO \parallel BC $. Let $ ABCD $ be a parallelogram, and $ M $ be the intersection point of its diagonals. Prove that $ AB = 2MO $.

2017 Zhautykov City MO grade IX P8
Given a triangle $ ABC $ with angles $ \angle A = 40 {} ^\circ $ and $ \angle B = 80 {} ^\circ $. On the segment $ AB $, the points $ K $ and $ L $ are taken (the point $ K $ lies between the points $ A $ and $ L $) such that $ AK = BL $ and $ \angle KCL = 30 {} ^\circ $ . Find the angle $ LCB $.

2018 Zhautykov City MO grade VII P4
Given a convex quadrilateral $ ABCD $, in which $ \angle B = \angle C = 90 ^\circ $, $ AB = BC = 2CD $. The point $ M $ is the midpoint of the side $ BC $, and $ N $ is the intersection point of the segments $ AC $ and $ BD $. Prove that the lines $ MN $ and $ AD $ intersect at a right angle.

In the figure below, the points $ A $ and $ B $ are marked on two parallel lines $ a $ and $ b $. It is known that angle $1$ is two times smaller than angle $2$, and angle $3$ is half that of angle $4$. Prove that angle $ ACB $ is half that of angle $ ADB $.

2018 Zhautykov City MO grade VIII P3
In the convex quadrilateral $ ABCD $ it is known: $ \angle ABC = 140 ^\circ $, $ \angle ADC = 115 ^ \circ $, $ \angle CBD = 40 ^ \circ $, $ \angle CDB = 65 ^\circ $. Calculate the angle $ \angle ACD $.

2018 Zhautykov City MO grade VIII P8 IX P7
Given three concentric circles of radii $3, 4$ and $5$. Intersecting chords $ AB $ and $ CD $ of a circle of radius $5$ ar tangent to circles of radii $3$ and $4$, respectively. Prove that the lines $ AC $ and $ BD $ intersect at a right angle.

2018 Zhautykov City MO grade IX P4
In the triangle $ ABC $ the point $ I $ is the center of the inscribed circle. On rays $ AI $ and $ BI $, points $ A_1 $ and $ B_1 $ respectively are taken for $ I $ and such that $ \angle ACA_1 = \angle BCB_1 = 90 ^\circ. $ Let $ M $ be the midpoint of the segment $ A_1B_1 $. Prove that the lines $ IM $ and $ AB $ are perpendicular.

2019 Zhautykov City MO grade VI P5
Cut the square into three parts, from which it would be possible to fold a triangle with three sharp corners and different sides. (You can cut it any way.)

2019 Zhautykov City MO grade VII P3
The bisector $ AD $ is drawn in the triangle $ ABC $. What is the angle of $ BAC, $ if the angle of $ B $ is two times the angle of $ C $ and $ CD = AD $?

2019 Zhautykov City MO grade VII P7
The bisector of angle $ BAC, $ of $ ABCD, $ intersects $ BC $ at $ M. $ Prove that $ AC = BC + BM. $

2019 Zhautykov City MO grade VIII P4
In the triangle $ ABC $, the sides $ AC $ and $ BC $ are equal. The bisector of the angle $ BAC $ intersects $ BC $ at the point $ E $. The point $ D $ is marked on the side of $ AB $. The lines $ AE $ and $ CD $ intersect at $ N $. It is known that $ \angle CDB = \angle CEA = 60 ^\circ $. Prove that the perimeter of the triangle $ CEN $ is equal to the segment $ AB $.

The diagonals of a convex cyclic quadrilateral $ ABCD $ intersect at a point $ O $. Let $ OA_1 $, $ OB_1 $, $ OC_1 $, $ OD_1 $ be the altitudes of the triangles $ OAB $, $ OBC $, $ OCD $, $ ODA $, respectively. It is known that $ A_1B_1 = 32 $, $ B_1C_1 = 23, $ $ C_1D_1 = 30 $. Find $ D_1A_1 $.


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