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Caucasus 2015-22 (Russia) 26p

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

Caucasus 2015 in English  here

collected inside aops:  Juniors  and Seniors
In 2016 it did not take place.

Juniors
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 $BP = CP$.

2017 Caucasus MO Juniors p4 /  Seniors p3
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.

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 $XYZ$.

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

Points $A'$ and $B'$ lie inside the parallelogram $ABCD$ and points $C'$ and $D'$ lie outside of it, so that all sides of 8-gon $AA'BB'CC'DD'$ are equal. Prove that  $A'$, $B'$, $C'$, $D'$ are concyclic.

In a triangle $ABC$ with  $\angle BAC = 90^{\circ}$ let $BL$ be the bisector, $L\in AC$. Let $D$ be a point symmetrical to $A$ with respect to $BL$. Let  $M$ be the circumcenter of $ADC$. Prove that $CM$, $DL$, and $AB$ are concurrent.


Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Let one of its internal tangents touches $\omega_1$ and $\omega_2$ at $A_1$ and $A_2$, respectively, and let one of its external tangents touches $\omega_1$ and $\omega_2$ at $B_1$ and $B_2$, respectively. Prove that if $A_1B_2 = A_2B_1$, then $A_1B_2 \perp A_2B_1$.

A regular triangle $ABC$ is given. Points $K$ and $N$ lie in the segment $AB$, a point $L$ lies in the segment $AC$, and a point $M$ lies in the segment $BC$ so that $CL=AK$, $CM=BN$, $ML=KN$. Prove that $KL \parallel MN$.

In a triangle $ABC$ let $K$ be a point on the median $BM$ such that $CK=CM$. It appears that
$\angle CBM = 2 \angle ABM$. Prove that $BC=MK$.

An acute triangle $ABC$ is given. Let $AD$ be its altitude, let $H$ and $O$ be its orthocenter and its
circumcenter, respectively. Let $K$ be the point on the segment $AH$ with $AK=HD$; let $L$ be the
point on the segment $CD$ with $CL=DB$. Prove that line $KL$ passes through $O$.

In parallelogram $ABCD$, points $E$ and $F$ on segments $AD$ and $CD$ are such that $\angle BCE=\angle BAF$. Points $K$ and $L$ on segments $AD$ and $CD$ are such that $AK=ED$ and $CL=FD$. Prove that $\angle BKD=\angle BLD$.
Point $P$ is chosen on the leg $CB$ of right triangle $ABC$ ($\angle ACB = 90^\circ$). The line $AP$ intersects the circumcircle of $ABC$ at point $Q$. Let $L$ be the midpoint of $PB$. Prove that $QL$ is tangent to a fixed circle independent of the choice of point $P$.

Seniors

2015 Caucasus MO grade 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 grade 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 p4 /  Seniors p3
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.

2018 Caucasus MO Seniors p2
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 p7
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 $AC$.

In a triangle $ABC$ let $I$ be the incenter. Prove that the circle passing through $A$ and touching $BI$ at $I$, and the circle passing through $B$ and touching $AI$ at $I$, intersect at a point on the circumcircle of $ABC$.

On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that  $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.

Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Let one of its internal tangents touches $\omega_1$ and $\omega_2$ at $A_1$ and $A_2$, respectively, and let one of its external tangents touches $\omega_1$ and $\omega_2$ at $B_1$ and $B_2$, respectively. Prove that if $A_1B_2 = A_2B_1$, then $A_1B_2 \perp A_2B_1$.

In $\triangle ABC$ with $AB\neq{AC}$ let $M$ be the midpoint of $AB$, let $K$ be the midpoint of the arc $BAC$ in the circumcircle of $\triangle ABC$, and let the perpendicular bisector of $AC$ meet the bisector of $\angle BAC$ at $P$ . Prove that $A, M, K, P$ are concyclic.

In an acute triangle $ABC$ let $AH_a$ and $BH_b$ be altitudes. Let $H_aH_b$ intersect the
circumcircle of $ABC$ at $P$ and $Q$. Let $A'$ be the reflection of $A$ in $BC$, and let $B'$ be the
reflection of $B$ in $CA$. Prove that $A', B'$, $P$, $Q$ are concyclic.

A triangle $\Delta$ with sidelengths $a\leq b\leq c$ is given. It appears that it is impossible to construct
a triangle from three segments whose lengths are equal to the altitudes of $\Delta$. Prove that $b^2>ac$.

4 tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral $P$, then
the following move could be performed: choose one of the tokens and shift it in the direction
perpendicular to the diagonal of $P$ not containing this token; while shifting tokens it is prohibited to
get three collinear tokens.
Suppose that initially tokens were at the vertices of a rectangle $\Pi$, and after a number of moves
tokens were at the vertices of one another rectangle $\Pi'$ such that $\Pi'$ is similar to $\Pi$ but not
equal to $\Pi $. Prove that $\Pi$ is a square.

Let $\omega$ is tangent to the sides of an acute angle with vertex $A$ at points $B$ and $C$. Let $D$ be an arbitrary point onn the major arc $BC$ of the circle $\omega$. Points $E$ and $F$ are chosen inside the angle $DAC$ so that quadrilaterals $ABDF$ and $ACED$ are inscribed and the points $A,E,F$ lie on the same straight line. Prove that lines $BE$ and $CF$ intersectat $\omega$.

Let $ABC$ be an acute triangle. Let $P$ be a point on the circle $(ABC)$, and $Q$ be a point on the segment $AC$ such that $AP\perp BC$ and $BQ\perp AC$. Lot $O$ be the circumcenter of triangle $APQ$. Find the angle $OBC$. 


source: http://cmo.adygmath.ru/

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