Kosovo TST 2015-20 7p (-18)

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

(only those not in IMO Shortlist)

2015 - 2020 (-18)

2015 Kosovo TST P5

In convex quadrilateral ABCD,diagonals AC and BD intersect at S and are perpendicular.

a)Prove that midpoints M,N,P,Q of AD,AB,BC,CD form a rectangular

b)If diagonals of MNPQ intersect O and AD=5,BC=10,AC=10,BD=11 find value of SO.

2016 Kosovo TST P5

Let be $ABC$ an acute triangle with $|AB|>|AC|$ . Let be $D$ point in side $AB$ such that $\angle ACD=\angle CBD$ . Let be $E$ the midpoint of segment $BD$ and $S$ let be the circumcenter of triangle $BCD$ . Show that points $A,E,S$ and $C$ lie on a circle .

2017 Kosovo TST P5

Given triangle $ABC$. Let $P$, $Q$, $R$, be the tangency points of inscribed circle of $\triangle ABC$ on sides $AB$, $BC$, $AC$ respectively. We take the reflection of these points with respect to midpoints of the sides they lie on, and denote them as $P',Q'$ and $R'$. Prove that $AP'$, $BQ'$, and $CR'$ are concurrent.

 

2018 tst missing

2018 Kosovo IMO training test p2

Given an acute triangle ${ABC}$, erect triangles ${ABD}$ and ${ACE}$ externally, so that ${\angle ADB= \angle AEC=90^o}$ and ${\angle BAD= \angle CAE}$. Let ${{A}_{1}}\in BC,{{B}_{1}}\in AC$ and ${{C}_{1}}\in AB$ be the feet of the altitudes of the triangle ${ABC}$, and let $K$ and $L$ be the midpoints of $B{{C}_{1}}$ and $CB_1$, respectively. Prove that the circumcenters of the triangles $AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}$ and ${DEA_1}$ are collinear.

2018 Kosovo IMO training test p4

It is given an acute triangle with $AB<AC$ and with circumcircle $\omega$. Let be point $D\in\omega$ such that $AD\bot BC$. Let be points $E$ and $F$ feet of altitudes from point $B$ and $C$, respectively. Let be points $P,Q$ and $R$ such that $BC\cap EF=P$ , $CD\cap BE=Q$ and $BD\cap CF=R$. Show that $P$ is midpoint of segment $QR$.

2019 Kosovo TST P4 (JBMO Shortlist 2018 G5)

Given a rectangle $ABCD$ such that $AB = b > 2a = BC$, let $E$ be the midpoint of $AD$. On a line parallel to $AB$ through point $E$, a point $G$ is chosen such that the area of $GCE$ is

$$(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)$$ Point $H$ is the foot of the perpendicular from $E$ to $GD$ and a point $I$ is taken on the diagonal $AC$ such that the triangles $ACE$ and $AEI$ are similar. The lines $BH$ and $IE$ intersect at $K$ and the lines $CA$ and $EH$ intersect at $J$. Prove that $KJ \perp AB$

2020 Kosovo TST P3

Let $ABCD$ be a cyclic quadrilateral with center $O$ such that $BD$ bisects $AC.$ Suppose that the angle bisector of $\angle ABC$ intersects the angle bisector of $\angle ADC$ at a single point $X$ different than $B$ and $D.$ Prove that the line passing through the circumcenters of triangles $XAC$ and $XBD$ bisects the segment $OX.$

by Viktor Ahmeti and Leart Ajvazaj, Kosovo