Euler Olympiad 2011-18 (Kazakhstan) 19p

geometry problems from Euler Olympiads (final stage, from Kazakstan)
with aops links in the names
a Junior Competition

2009 - 2018

2009 Euler Olympiad P3
In the triangle $ ABC $, the sides $ AB $ and $ BC $ are equal. The point $ D $ inside the triangle is such that the angle $ ADC $ is twice as large as the angle $ ABC $. Prove that the double distance from the point $ B $ to the line bisecting the angles externally to the angle $ ADC $ is $ AD + DC $.

2009 Euler Olympiad P6
In the convex quadrilateral $ ABCD $ the relations $ AB = BD $ are satisfied; $ \angle ABD = \angle DBC $. On the diagonal $ BD $ there was a point $ K $ such that $ BK = BC $. Prove that $ \angle KAD = \angle KCD $.

2010 Euler Olympiad P3
In the quadrilateral $ ABCD $, the side $ AB $ is equal to the diagonal $ AC $ and is perpendicular to the side $ AD $, and the diagonal $ AC $ is perpendicular to the side $ CD $. On the side $ AD $, a point $ K $ is taken such that $ AC = AK $. The bisector of the angle $ ADC $ intersects $ BK $ at the point $ M $. Find the angle $ ACM $.

2010 Euler Olympiad P6
In the convex quadrilateral $ ABCD $, the angles $ B $ and $ D $ are equal, $ CD = 4BC $, and the bisector of the angle $ A $ passes through the middle of the side $ CD $. What can the $ AD / AB $ ratio be?

2011 Euler Olympiad P4
Inside the convex quadrilateral $ ABCD $, in which $ AB = CD $, the point $ P $ is chosen in such a way that the sum of the angles $ PBA $ and $ PCD $ is $180$ degrees. Prove that $PB+PC<AD$.

2011 Euler Olympiad P6
The convex pentagon $ABCDE$ is such that $ AB \parallel CD $, $ BC \parallel AD $, $ AC \parallel DE $, $ CE \perp BC $. Prove that $EC$ is the bisector of the angle $BED$.

2012 Euler Olympiad P1
On the side $ BC $ of the triangle $ ABC $ the point $ D $ is taken in such a way that the perpendicular bisector of segment $ AD $ passes through the center of the circle inscribed in the triangle $ ABC $. Prove that this perpendicular passes through a vertice of the triangle $ ABC $.

2012 Euler Olympiad P7
The angles of the triangle $ABC$ satisfy the condition $2 \angle A + \angle B = \angle C$. Inside this triangle, the point $ K $ is chosen on the bisector of the angle $ A $ such that $BK = BC$. Prove that $\angle KBC = 2 \angle KBA$.

2013 Euler Olympiad P3
The diagonals of the convex quadrilateral $ ABCD $ are equal and intersect at the point $ O $. The point $ P $ inside the triangle $ AOD $ is such that $ CD \parallel BP $ and $ AB \parallel CP $. Prove that the point $ P $ lies on the bisector of the angle $ AOD $.

2013 Euler Olympiad P6
In the convex quadrilateral $ ABCD $ in which $ AB = CD $, the points $ K $ and $ M $ are chosen on the sides $ AB $ and $ CD $, respectively. It turned out that $ AM = KC $, $ BM = KD $. Prove that the angle between the lines $ AB $ and $ KM $ is equal to the angle between the lines $ KM $ and $ CD $.

On the side  $AB$ of a  triangle  $ABC$ with an angle of  $100 ^\circ$  at the vertice $C$,  points  $P$ and $Q$ such that  $AP = BC$ and $BQ = AC$ . Let $M$, $N$, $K$ be  midpoints of $AB$, $CP$, $CQ$, respectively. Find the angle $NMK$.

In the triangle $ABC$  , side  $AB$ is greater than side $BC$. On the extension of the side $BC$ beyond  the point  $C$,  noted the point $N$  so that  $2BN = AB+BC$. Let  $BS$ be the angle bisector of $ABC$, $M$ be the midpoint of the side $AC$, а $L$  be a point on the side $BS$  such that $ML \parallel AB$ . Prove that $2LN = AC$.

2015 Euler Olympiad P8
Let $CK$  be angle bisector of the triangle $ABC$. On the sides $BC$ and $AC$, are chosen points $L$ and $T$ respectively, such that $CT = BL$ and  $TL = BK$. Prove that the triangle $LTC$ is similar to the original one.

An equilateral triangle $ABC$. is given. Point $D$  is chosen on the extension of the side $AB$ beyond the point $A$, point $E$  on the extension of $BC$ beyond the point $C$, and point $F$ on the extension of $AC$ beyond the point $C$ so that  $CF = AD$ and $AC+EF = DE$. Find the angle $BDE$.

2016 Euler Olympiad P8
A parallelogram $ABCD$. is given. On the sides $AB$ and  $BC$ and the extension of the side  $CD$  beyond the point  $D$ , the points $K$, $L$ and $M$ are chosen, and so that the triangles $KLM$ and  $BCA$  are congruent  (precisely with such a correspondence of the vertices). The segment $KM$ intersects the segment $AD$ at the point $N$. Prove that $LN  \parallel  AB$.

Diagonals of a convex quadrilateral $ABCD$ intersect at a point $E$. It is known that $AB=BC=CD=DE=1$. Prove that $AD<2$

2017 Euler Olympiad P6
In a convex quadrilateral $ABCD$ angles $A$ and  $C$ are equal  $100^\circ$. On the sides $AB$ and $BC$ points $X$ and  $Y$ are selected  so that $AX = CY$ . It turned out that the line $YD$ is parallel to the angle bisector of the angle $ABC$. Find the angle $AXY$.

2018 Euler Olympiad P3
Diagonals of a convex quadrilateral   $ABCD$  are equal and intersect at a point $K$ . Inside the triangles $AKD$ and  $BKC$ select points  $P$ and  $Q$ so that  $\angle KAP = \angle KDP = \angle KBQ = \angle KCQ.$  Prove that line  $PQ$ is parallel to the bisector of the angle $AKD$.

2018 Euler Olympiad P8
Vertex  $F$ of a parallelogram $ACEF$ lies on the side  $BC$ of a parallelogram $ABCD$ . It is known that $AC = AD$ and $AE = 2CD$. Prove that $\angle CDE = \angle BEF.$


source: matol.kz/nodes/74

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