Serbia TST (IMO - EGMO - RMM) 2003-19 27p

geometry problems from Serbian Team Selection Tests (TST)
with aops links in the names

(only those not in IMO Shortlist)

IMO TST 2003 - 2019

2003 Serbia TST P2
Let $M$ and $N$ be the distinct points in the plane of the triangle $ABC$ such that $AM : BM : CM = AN : BN : CN$. Prove that the line $MN$ contains the circumcenter of $\triangle ABC$.

2004 Serbia TST P1
Let $ABCD$ be a square and $K$ be a circle with diameter $AB$. For an arbitrary point $P$ on side $CD$, segments $AP$ and $BP$ meet $K$ again at points $M$ and $N$, respectively, and lines $DM$ and $CN$ meet at point $Q$. Prove that $Q$ lies on the circle $K$ and that $AQ : QB = DP : PC$.

2006 Serbia TST P2
A point $P$ is taken in the interior of a right triangle$ ABC$ with $\angle C = 90$ such hat
$AP = 4, BP = 2$, and$ CP = 1$. Point $Q$ symmetric to $P$ with respect to $AC$ lies on
the circumcircle of triangle $ABC$. Find the angles of triangle $ABC$.

2009 Serbia TST P1
Let $ \alpha$ and $ \beta$ be the angles of a non-isosceles triangle $ ABC$ at points $ A$ and $ B$, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in $ D$ and $ E$, respectively. Prove that the acute angle between the lines $ DE$ and $ AB$ isn't greater than $ \frac{|\alpha-\beta|}3$.

2009 Serbia TST P6
Let $ k$ be the inscribed circle of non-isosceles triangle $ \triangle ABC$, which center is $ S$. Circle $ k$ touches sides $ BC,CA,AB$ in points $ P,Q,R$ respectively. Line $ QR$ intersects $ BC$ in point $ M$. Let a circle which contains points $ B$ and $ C$ touch $ k$ in point $ N$. Circumscribed circle of $ \triangle MNP$ intersects line $ AP$ in point $ L$, different from $ P$. Prove that points $ S,L$ and $ M$ are collinear.

2012 Serbia TST P3 
Let $P$ and $Q$ be points inside triangle $ABC$ satisfying $\angle PAC=\angle QAB$ and $\angle PBC=\angle QBA$.
a) Prove that feet of perpendiculars from $P$ and $Q$ on the sides of triangle $ABC$ are concyclic.
b) Let $D$ and $E$ be feet of perpendiculars from $P$ on the lines $BC$ and $AC$ and $F$ foot of perpendicular from $Q$ on $AB$. Let $M$ be intersection point of $DE$ and $AB$. Prove that $MP\bot CF$.

2013 Serbia TST P2
In an acute $\triangle ABC$ ($AB \neq AC$) with angle $\alpha$ at the vertex $A$, point $E$ is the nine-point center, and $P$ a point on the segment $AE$. If $\angle ABP = \angle ACP = x$, prove that $x = 90$° $ -2 \alpha $.

 by Dusan Djukic
2017 Serbia TST P1
Let $ABC$ be a triangle and $D$ the midpoint of the side $BC$. Define points $E$ and $F$ on $AC$ and $B$, respectively, such that $DE=DF$ and $\angle EDF =\angle BAC$. Prove that $$DE\geq \frac {AB+AC} 4.$

Let $H $ be the orthocenter of $ABC $ ,$AB\neq AC $ ,and let $F $ be a point on circumcircle of $ABC $ such that $\angle AFH=90^{\circ} $.$K $ is the symmetric point of $H $ wrt $B $.Let $P $ be a point such that $\angle PHB=\angle PBC=90^{\circ} $,and $Q $ is the foot of $B $ to $CP $.Prove that $HQ $ is tangent to tge circumcircle of $FHK$.

2019 Serbia TST P2
Given triangle $\triangle ABC $ with $AC\neq BC $,and let $D $ be a point inside triangle such that $\measuredangle ADB=90^{\circ} + \frac {1}{2}\measuredangle ACB $.Tangents from $C $ to the circumcircles of $\triangle ABC $ and $\triangle ADC $ intersect $AB $ and $AD $ at $P $ and $Q $ , respectively.Prove that $PQ $ bisects the angle $\measuredangle BPC $.

EGMO TST 2012-19

2012 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 $\omega$.

2012 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  $\rho = r - r'$.

2013 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 $\angle BAC = 120^o$.

2013 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 $f (P) =\frac{PA + PB}{PC + PD}$

2014 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 $AC$.

2015 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 $IM$.

2016 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 $\angle AFE=\angle CDF$.

2017 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 $\odot ABC$.

2018 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.

On the sides $BC, CA$ and $AB$ of triangle $ABC$ are given the points $P, Q$ and $R$ respectively so the quadrilateral $AQPR$ is cyclic and $BR = CQ$. Tangents to the circumscribed the circle of the triangle $ABC$ at points $B$ and $C$ intersect the $PR$ and $PQ$ at points $X$ and $Y$, in a row. Prove that $PX = PY$.

RMM TST 2012-19

2012 Serbia RMM TST P2
Inside the convex quadrilateral $ABCD$ (not a trapezoid)  given is a point $X$ such that $\angle ADX =  \angle BCX <90^o$ and $\angle DAX =\angle CBX <90^o$. If the perpendicular bisectors of $AB$ and $CD$ intersect at point $Y$, prove that $\angle AYB = 2 \angle ADX$

2015 Serbia RMM TST P2
Given an acute isosceles triangle $ABC$ with  $AB = AC$. Point $M$ is the midpoint of the arc $BC$ (not containing $A$) of the circumcscribed circle of triangle $ABC$. The line passing through $M$ parallel to $AC$ intersects $BC$ and $AB$ at points $D$ and $E$, respectively. The line passing through $D$ parallel to $AB$ intersects AC at point F. Prove that $\angle M EF = 90^o$

Given a pentagon with a broken line connecting two vertices and dividing into two congruent pentagons. Prove that the starting pentagon has a pair of equal angles, and that the pentagons intto which it is divided have two pairs of parallel sides.

Given a circle $k$ with center at point $O$ and point $A \in  k$. Let $C$ be the symmetric point of $O$ wrt $A$ and $B$ be the midpoint of $AC$. A point $Q \in  k$ such that $\angle AOQ$ is obtuse. Let  the line $QO$ and the perpendicular bisector of $CQ$ intersect at point $P$. Prove $\angle P OB = 2\angle P BO$.

2018 Serbia RMM TST P2
Let $P$ be a point inside triangle $ABC$ such that $\angle AP B \ne 90^o$.  Lines $AP$ and $BP$ intersect the sides $BC$ and $AC$ respectively at points $D$ and $E$. Let $K$ and $L$ be the orthocentres of triangles $AP E$ and $BPD$. Prove that point $C$ lies on the line $KL$ if and only if the angle $ACB$ is right.

2019 Serbia RMM TST P4 (Brazil MO 2015 P6)
In a scalene triangle $ABC$, $H$ is the orthocenter and $G$ is the centroid. The points $X, Y$ and $Z$ are selected respectively on sides $BC, CA$ and $AB$ so that $\angle AXB =\angle BY C = \angle CZA$.  The circumscribed circles of triangles $BXZ$ and $CXY$ intersect at $P \ne  X$. Prove that $\angle GP H = 90^o$.

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