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\$.