Serbia TST 2003-18 10p

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

(only those not in IMO Shortlist)

2003 - 2018
(in most years TST did not take place)

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 $FHK$.

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