EGMO TST 31p

geometry problems from Team Selection Tests for European's Girl Math Olympiad (EGMO TST)
with aops links in the names
                                  
                                      Under Construction


Bosnia and Herzegoniva 2017-18


2017 Bosnia and Herzegovina  EGMO TST P2
It is given triangle $ABC$ and points $P$ and $Q$ on sides $AB$ and $AC$, respectively, such that $PQ\mid\mid BC$. Let $X$ and $Y$ be intersection points of lines $BQ$ and $CP$ with circumcircle $k$ of triangle $APQ$, and $D$ and $E$ intersection points of lines $AX$ and $AY$ with side $BC$. If $2\cdot DE=BC$, prove that circle $k$ contains intersection point of angle bisector of $\angle BAC$ with $BC$ 

2018 Bosnia and Herzegovina  EGMO TST P3
Let $O$ be a circumcenter of acute triangle $ABC$ and let $O_1$ and $O_2$ be circumcenters of triangles $OAB$ and $OAC$, respectively. Circumcircles of triangles $OAB$ and $OAC$ intersect side $BC$ in points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Perpendicular bisector of side $BC$ intersects side $AC$ in point $F$($F \neq A$). Prove that circumcenter of triangle $ADE$ lies on $AC$ iff $F$ lies on line $O_1O_2$

Netherlands 2012-15

only the years 2012-2015 the BxMO TST counted also for EGMO, 
the following years no extra TST took place for EGMO, while the BxMO keeps on happening


2012 Moldova BxMO / EGMO TST P2
Let  $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of  $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.

2012 Moldova BxMO / EGMO TST P4
Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$. Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.


In quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.

Let $ABCD$ be a cyclic quadrilateral for which $|AD| =|BD|$. Let $M$ be the intersection of $AC$ and $BD$. Let $I$ be the incentre (centre of the inscribed circle) of $\triangle BCM$. Let $N$ be the second point of intersection of $AC$ and the circumscribed circle of $\triangle BMI$. Prove that $|AN| \cdot |NC| = |CD | \cdot  |BN|$.

2014 Moldova BxMO / EGMO TST P3
In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

In a triangle $ABC$ the point $D$ is the intersection of the interior angle bisector of  $\angle BAC$ and side $BC$. Let $P$ be the second intersection point of the exterior angle bisector of $\angle BAC$ with the circumcircle of $\angle ABC$. A circle through $A$ and $P$ intersects line segment $BP$ internally in $E$ and line segment $CP$ internally in $F$. Prove that $\angle DEP = \angle  DFP$.

Moldova 2017-18 

Let us denote the midpoint of $AB$ with $O$. The point $C$, different from $A$ and $B$ is on the circle $\Omega$ with center $O$ and radius $OA$ and the point $D$ is the foot of the perpendicular from $C$ to $AB$. The circle with center $C$ and radius $CD$ and $\omega$ intersect at $M$, $N$. Prove that $MN$ cuts $CD$ in two equal segments.

The points $P$ and $Q$ are placed in the interior of the triangle $\Delta ABC$ such that $m(\angle PAB)=m(\angle QAC)<\frac{1}{2}m(\angle BAC)$ and similarly for the other $2$ vertices($P$ and $Q$ are isogonal conjugates). Let $P_{A}$ and $Q_{A}$ be the intersection points of $AP$ and $AQ$ with the circumcircle of $CPB$, respectively $CQB$. Similarly the pairs of points $(P_{B},Q_{B})$ and $(P_{C},Q_{C})$ are defined. Let $PQ_{A}\cap QP_{A}=\{M_{A}\}$, $PQ_{B}\cap QP_{B}=\{M_{B}\}$, $PQ_{C}\cap QP_{C}=\{M_{C}\}$.
Prove the following statements:
$1.$ Lines $AM_{A}$, $BM_{B}$, $CM_{C}$ concur.
$2. $ $M_{A}\in BC$, $M_{B}\in CA$, $M_{C}\in AB$

Let $\triangle ABC $ be an acute triangle.$O$ denote its circumcenter.Points $D$,$E$,$F$ are the midpoints of the sides $BC$,$CA$,and $AB$.Let $M$ be a point on the side $BC$  .  $ AM \cap EF = \big\{ N \big\} $ .  $ON \cap \big( ODM \big) = \big\{ P  \big\} $ Prove that $M'$ lie on $\big(DEF\big)$ where $M'$ is the symmetrical point of $M$ thought the midpoint of $DP$.

Let $ABCD$ be a isosceles trapezoid with $AB  \| CD $ , $AD=BC$, $ AC \cap BD = $ { $O$ }. $ M $ is the midpoint of the side $AD$ . The circumcircle of triangle $ BCM $ intersects again the side $AD$ in $K$. Prove that $OK  \|  AB $ . 

Serbia 2011-17  [x refers to egmo x+1]

2011 Serbia EGMO TST P3
Let $\omega$ be the circumcircle of an acute angled triangle $ABC$. On sides $AB$ and $AC$ of this triangle were select  $E$ and $D$, respectively, so that $\angle ABD = \angle ACE$. Lines $BD$ and $CE$ intersect the circle $\omega$ at $M$ and $N$ ($M \ne B$ and $N \ne C$), respectively. The tangents at the points $B$ and $C$ on the circle $\omega$, intersect the line  $DE$ at  $P$ and $Q$ respectively.  Prove that the intersection point of lines $PN$ and $QM$ lies on the circle $\omega$.

2011 Serbia EGMO TST P7
Let the $r$ of the radius of the incircle of the triangle $ABC$. Let  $D, E$, and $F$ be points on the sides $BC, CA$, and $AB$  respectively, such that the  incircles of the triangles $AEF, BFD$ and $CDE$ have equal radius  $\rho$. If $r '$ is the radius of the incircle of the triangle $DEF$, prove that it is  $\rho = r - r'$.

2012 Serbia EGMO TST P2
Let $O$ be the center of the circumcircle , and $AD$ ($D \in BC$) be the interior angle bisector of a triangle ABC. Let $\ell$ be a line passing through $O$ parallel to  $AD$. Prove that $\ell$  passes through the orthocenter of the triangle $ABC$ if and only if $\angle BAC = 120^o$.

2012 Serbia EGMO TST P4
Let $ABCD$ be a square of the plane $P$. Define the minimum and the maximum the value of the function $f: P \to R$ is given by $f (P) =\frac{PA + PB}{PC + PD}$

2013 Serbia EGMO TST P1
Let $ABC$ be a triangle. Circle $k_1$ passes through $A$ and $B$ and touches the line $AC$, and circle $k_2$ passes through $C$ and $A$ and touches the line $AB$. Circle $k_1$ intersects line $BC$ at point  $D$ ($D \ne B$) and also  intersects the circle $k_2$ at point $E$ ($E \ne A$). Prove that the line $DE$ bisects segment $AC$.

2014 Serbia EGMO TST P3
The incircle of the triangle $ABC$ touches the side $BC$ at point $D$. Let $I$ be the center of the incircle $k$, $M$ the midpoint of side $BC$, and $K$ the  orthocenter of triangle $AIB$. Prove that $KD$ is perpendicular to $IM$.

2015 Serbia EGMO TST P3
Let $ABCD$ be cyclic quadriateral and let $AC$ and $BD$ intersect at $E$ and $AB$ and $CD$ at $F$. Let $K$ be point in plane such that $ABKC$ is parallelogram. Prove $\angle AFE=\angle CDF$.

2016 Serbia EGMO TST P3
In $\triangle ABC$ the incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$, respectively. Let $G$ be the foot of the altitude from $D$ in $\triangle DEF$. Prove that one of intersections of line $IG$ and $\odot ABC$ is the antipode of $A$ in $\odot ABC$.

2017 Serbia EGMO TST P4
A parallelogram $ABCD$  is given. On sides $AB,  BC$ and the extension of side $CD$ behind the vertice $D$ , lie the points $ K, L$ and $M$  respectively, such that $KL = BC, LM = CA$ and $MK = AB$. $KM$ intersects $AD$ at point $N$ .Prove that $LN  // AB$.

Let $ABCD$ be an isosceles trapezoid ($AB // CD$) . Let $E$ be  a point of that arc of $AB$ of the circumcircle of the trapezoid,  that does not contain the trapezoid. From each of $A$ and $B$, we drop perpendicular on $EC$ and $ED$. Prove that the those four projections of $A$ and $B$ on $EC$ and $ED$ are concyclic.

Turkey 2013-18 

2013 Turkey EGMO TST P3
Altitudes $AD$ and $CE$ of an acute angled triangle $ABC$ intersect at point $H$  . Let $K$ be the midpoint of  side $AC$ and $P$ be the  midpoint of segment $DE$  . Let $Q$ be the symmetrical point of $K$ wrt line $AD$.  Prove that $\angle QPH   = 90^o$

2013 Turkey EGMO TST P5
In a triangle $ABC$, $AB=AC$. A circle passing through $A$ and $C$ intersects $AB$ at $D$. Angle bisector from $A$ intersects the circle at $E$. (different from $A$) Prove that orthocenter of $AEB$ is on this circle.

2014 Turkey EGMO TST P1
Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and $AD$ intersect the circumcircle of $ABC$ for the second time at $E$. Let $P$ be the point symmetric to the point $E$ with respect to the point $D$ and $Q$ be the point of intersection of the lines $CP$ and $AB$. Prove that if $A,C,D,Q$ are concyclic, then the lines $BP$ and $AC$ are perpendicular.

2014 Turkey EGMO TST P5
Let $ABC$ be a triangle with circumcircle $\omega$ and let $\omega_A$ be a circle drawn outside $ABC$ and tangent to side $BC$ at $A_1$ and tangent to $\omega$ at $A_2$. Let the circles $\omega_B$ and $\omega_C$ and the points $B_1, B_2, C_1, C_2$ are defined similarly. Prove that if the lines $AA_1, BB_1, CC_1$ are concurrent, then the lines $AA_2, BB_2, CC_2$ are also concurrent.

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and $P$ be a point inside the $ABD$ satisfying $\angle PAD=90^\circ - \angle PBD=\angle CAD$. Prove that $\angle PQB=\angle BAC$, where $Q$ is the intersection point of the lines $PC$ and $AD$.

2016 Turkey EGMO TST P3
Let $X$ be a variable point on the side $BC$ of a triangle $ABC$. Let $B'$ and $C'$ be points on the rays $[XB$ and $[XC$, respectively, satisfying $B'X=BC=C'X$. The line passing through $X$ and parallel to $AB'$ cuts the line $AC$ at $Y$ and the line passing through $X$ and parallel to $AC'$ cuts the line $AB$ at $Z$. Prove that all lines $YZ$ pass through a fixed point as $X$ varies on the line segment $BC$.

2016 Turkey EGMO TST P4
In a convex pentagon, let the perpendicular line from a vertex to the opposite side be called an altitude. Prove that if four of the altitudes are concurrent at a point then the fifth altitude also passes through this point.

2017 Turkey EGMO TST P4
On the inside of the triangle $ABC$ a point $P$ is chosen with $\angle BAP = \angle CAP$. If $\left | AB \right |\cdot \left | CP \right |= \left | AC \right |\cdot \left | BP \right |= \left | BC \right |\cdot \left | AP \right |$ , find all possible values of the angle $\angle ABP$.

2018 Turkey EGMO TST P1
Let $ABCD$ be a cyclic quadrilateral and $w$ be its circumcircle.  For a given point $E$ inside $w$, $DE$ intersects $AB$ at $F$ inside $w$. Let $l$ be a line passes through $E$ and tangent to circle $AEF$. Let $G$ be any point on $l$ and inside the quadrilateral $ABCD$. Show that if $\angle GAD =\angle BAE$ and $\angle GCB + \angle GAB = \angle EAD + \angle AGD +  \angle ABE$ then $BC$, $AD$ and $EG$ are concurrent.


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