Turkey JBMO TST 2012-22 15p (-19,-20)

geometry problems from Turkish Junior Balkan Mathematical Olympiads Team Selection Tests (JBMO TST) with aops links

collected inside aops here

2012-18 , 2022
(2017 got cancelled)

2012 Turkey JBMO TST P1
Let $[AB]$ be a chord of the circle $\Gamma$ not passing through its center and let $M$ be the midpoint of $[AB].$ Let $C$ be a variable point on $\Gamma$ different from $A$ and $B$ and $P$ be the point of intersection of the tangent lines at $A$ of circumcircle of $CAM$ and at $B$ of circumcircle of $CBM.$ Show that all $CP$ lines pass through a fixed point.

2013 Turkey JBMO TST P1

Let $D$ be a point on the side $BC$ of an equilateral triangle $ABC$ where $D$ is different than the vertices. Let $I$ be the excenter of the triangle $ABD$ opposite to the side $AB$ and $J$ be the excenter of the triangle $ACD$ opposite to the side $AC$. Let $E$ be the second intersection point of the circumcircles of triangles $AIB$ and $AJC$. Prove that $A$ is the incenter of the triangle $IEJ$.

2013 Turkey JBMO TST P7
In a convex quadrilateral $ABCD$ diagonals intersect at $E$ and $BE = \sqrt{2}\cdot ED, \: \angle BEC = 45^{\circ}.$ Let $F$ be the foot of the perpendicular from $A$ to $BC$ and $P$ be the second intersection point of the circumcircle of triangle $BFD$ and line segment $DC$. Find $\angle APD$.

2014 Turkey JBMO TST P1
In a triangle $ABC$, the external bisector of $\angle BAC$ intersects the ray $BC$ at $D$. The feet of the perpendiculars from $B$ and $C$ to line $AD$
are $E$ and $F$, respectively and the foot of the perpendicular from $D$ to $AC$ is $G$. Show that $\angle DGE +  \angle DGF = 180^{\circ}$.

2014 Turkey JBMO TST P7
Let a line $\ell$ intersect the line $AB$ at $F$, the sides $AC$ and $BC$ of a triangle $ABC$ at $D$ and $E$, respectively and the internal bisector of the angle $BAC$ at $P$. Suppose that $F$ is at the opposite side of $A$ with respect to the line $BC$, $CD = CE$ and $P$ is in the interior the triangle $ABC$. Prove that $FB \cdot FA+CP^2 = CF^2 \iff AD \cdot BE = PD^2.$

2015 Turkey JBMO TST P2
Let $ABCD$ be a convex quadrilateral and let $\omega$ be a circle tangent to the lines $AB$ and $BC$ at points $A$ and $C$, respectively. $\omega$ intersects the line segments $AD$ and $CD$ again at $E$ and $F$, respectively, which are both different from $D$. Let $G$ be the point of intersection of the lines $AF$ and $CE$. Given $\angle ACB=\angle GDC+\angle ACE$, prove that the line $AD$ is tangent to th circumcircle of the triangle $AGB$

2015 Turkey JBMO TST P6
Find the greatest possible integer value of the side length of an equilateral triangle whose vertices belong to the interior region of a square with side length $100$.

2016 Turkey JBMO TST P4
In a trapezoid $ABCD$ with $AB<CD$ and $AB \parallel CD$, the diagonals intersect each other at $E$. Let $F$ be the midpoint of the arc $BC$ (not containing the point $E$) of the circumcircle of the triangle $EBC$. The lines $EF$ and $BC$ intersect at $G$. The circumcircle of the triangle $BFD$ intersects the ray $[DA$ at $H$ such that $A \in [HD]$. The circumcircle of the triangle $AHB$ intersects the lines $AC$ and $BD$ at $M$ and $N$, respectively. $BM$ intersects $GH$ at $P$, $GN$ intersects $AC$ at $Q$. Prove that the points $P, Q, D$ are collinear.

2016 Turkey JBMO TST P5
In an acute triangle $ABC$, the feet of the perpendiculars from $A$ and $C$ to the opposite sides are $D$ and $E$, respectively. The line passing through $E$ and parallel to $BC$ intersects $AC$ at $F$, the line passing through $D$ and parallel to $AB$ intersects $AC$ at $G$. The feet of the perpendiculars from $F$ to $DG$ and $GE$ are $K$ and $L$, respectively. $KL$ intersects $ED$ at $M$. Prove that $FM \perp ED$.

2017 Turkey JBMO TST  got cancelled.

2018 Turkey JBMO TST P3
Let $H$ be the orthocenter of an acute angled triangle $ABC$. Circumcircle of the triangle $ABC$ and the circle of diameter $[AH]$ intersect at point $E$, different from $A$. Let $M$ be the midpoint  of the small arc $BC$ of the circumcircle of the triangle $ABC$ and let $N$ the midpoint of the large arc $BC$ of the circumcircle of the triangle $BHC$  Prove that  points $E, H, M, N$ are concyclic.

2018 Turkey JBMO TST P6
A point $E$ is located inside a parallelogram  $ABCD$  such that $\angle BAE = \angle BCE$. The centers of the circumcircles of the triangles $ABE,ECB, CDE$ and $DAE$ are concyclic.

2019, 2020 unknown

2021 Turkey JBMO TST P1

In an acute-angled triangle $ABC$, the circle with diameter $[AB]$ intersects the altitude drawn from vertex $C$ at a point $D$ and the circle with diameter $[AC]$ intersects the altitude drawn from vertex $B$ at a point $E$. Let the lines $BD$ and $CE$ intersect at $F$. Prove that $$AF\perp DE$$

2021 Turkey JBMO TST P8

Circles $w_1$ and $w_2$ have different diameters and externally tangent to each other at $X$. Points $A$ and $B$ are on $w_1$, points $C$ and $D$ are on $w_2$ such that $AC$ and $BD$ are common tangent lines of these two circles. $CX$ intersects $AB$ at $E$ and $w_1$ at $F$ second time. $(EFB)$ intersects $AF$ at $G$ second time. If $AX \cap CD =H$, show that points $E, G, H$ are collinear.

2022 Turkey JBMO TST P4

Given a convex quadrilateral $ABCD$ such that $m(\widehat{ABC})=m(\widehat{BCD})$. The lines $AD$ and $BC$ intersect at a point $P$ and the line passing through $P$ which is parallel to $AB$, intersects $BD$ at $T$. Prove that

$$m(\widehat{ACB})=m(\widehat{PCT})$$

2022 Turkey JBMO TST P7

In a triangle $ABC$ such that $\widehat{B}<\widehat{C}$, let $K$ be the center of the excircle that is tangent to the side $[AC]$. The lines $AK$ and $BC$ intersect at $D$, and $E$ is the center of the circumcircle of $BKC$. Prove that

$$\frac 1{|KA|}=\frac 1{|KD|}+\frac 1{|KE|}$$

source:
geomania.org/forum/index.php?action=forum#c12