Caucasus 2015-18 10p

geometry problems from Caucasus Mathematical Olympiads
 with aops links in the names 

2015 - 2018

In the convex quadrilateral $ABCD$, point $K$ is the midpoint of $AB$, point $L$ is the midpoint of $BC$, point $M$ is the midpoint of CD, and point $N$ is the midpoint of $DA$. Let $S$ be a point lying inside the quadrilateral $ABCD$ such that $KS = LS$ and  $NS = MS$ .Prove that $\angle KSN =  \angle MSL$.

Let $AL$ be the angle bisector of the acute-angled triangle $ABC$. and $\omega$ be the circle circumscribed about it. Denote by $P$ the intersection point of the extension of the altitude $BH$ of the triangle $ABC$ with the circle $\omega$ . Prove that if $\angle BLA= \angle BAC$, then $BP = CP$.

2015 Caucasus MO X P5
Let $AA_1$ and $CC_1$ be the altitudes of the acute-angled  triangle $ABC$. Let $K,L$ and $M$ be the midpoints of the sides $AB,BC$ and $CA$ respectively. Prove that if  $\angle C_1MA_1 =\angle ABC$, then $C_1 K = A_1L$.

2015 Caucasus MO XI P4
The midpoint of the edge $SA$ of the triangular pyramid of $SABC$ has equal distances  from all the vertices of the pyramid. Let $SH$ be the height of the pyramid. Prove that $BA^2 + BH^2 = C A^2 + CH^2$.

In 2016 it did not take place.

2017 Caucasus MO Juniors problem 4 /  Seniors problem 3
In an acute traingle $ABC$ with  $AB< BC$ let $BH_b$ be its altitude, and let $O$ be the circumcenter. A line through $H_b$ parallel to $CO$ meets $BO$ at $X$. Prove that $X$ and the midpoints of $AB$ and $AC$ are collinear.

2017 Caucasus MO Juniors problem 6
A triangle is cut by $3$ cevians from its $3$ vertices into $7$ pieces: $4$ triangles and $3$ quadrilaterals. Determine if it is possible that all $3$ quadrilaterals are inscribed. 
By centroid  of a quadrilateral $PQRS$ we call a common point of two lines through the midpoints of its opposite sides. Suppose that $ABCDEF$ is a hexagon inscribed into the circle $\Omega$ centered at $O$. Let  $AB=DE$, and  $BC=EF$. Let $X$, $Y$, and $Z$ be centroids of $ABDE$,  $BCEF$; and $CDFA$, respectively. Prove that $O$ is the orthocenter of triangle $XYZ$.

2018 Caucasus MO Juniors problem 6

Given a convex quadrilateral $ABCD$ with $\angle BCD=90^\circ$. Let $E$ be the midpoint of $AB$. Prove that $2EC \leq  AD+BD$.

2018 Caucasus MO Seniors problem 2
Let $I$ be the incenter of an acute-angled triangle $ABC$. Let $P$, $Q$, $R$ be points on sides $AB$, $BC$, $CA$ respectively, such that $AP=AR$, $BP=BQ$ and $\angle PIQ = \angle BAC$. Prove that $QR \perp AC$.

2018 Caucasus MO Seniors problem 7
In an acute-angled triangle $ABC$, the altitudes from $A,B,C$ meet the sides of $ABC$ at $A_1$, $B_1$, $C_1$, and meet the circumcircle of $ABC$ at $A_2$, $B_2$, $C_2$, respectively. Line  $A_1 C_1$ intersects the circumcircles of triangles $AC_1 C_2$ and $CA_1 A_2$ at points $P$ and $Q$ ($Q\neq A_1$, $P\neq C_1$). Prove that the circle $PQB_1$ touches the line $AC$.

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