### Greece JBMO TST 1998 - 2019 22p

geometry problems from Greek Junior Balkan Mathematical Olympiads Team Selection Tests (JBMO TST)

1998 - 2019

1998 Greece JBMO TST P3
Let $ABCD$ be a trapezoid ($AB//CD$) and points $M,N$ of sides $AD,BC$ respectively such that $MN//AB$. Prove that $DC \cdot MA +AB \cdot MD = MN \cdot AD$.

$\Phi$ is the union of all triangles that are symmeeric of the triangle $ABC$ wrt a point $O$, $O$ lies in a side of the triangle. If the  area of the triangle is $E$, find the area of $\Phi$.

2000 Greece JBMO TST P2
Let $ABCD$ be a convex quadrilateral with $AB=CD$. From a random  point $P$ of it's diagonal $BD$, we draw a line parallel to $AB$ that intersects $AD$ at point $M$ and a line parallel to $CD$ that intersects $BC$ at point $N$. Prove that:
a) The sum $PM+PN$ is constant, independent of the position of $P$ on the diagonal $BD$.
b) $MN\le BD$. When the equality holds?

2001 Greece JBMO TST P2
Let ABCD be a quadrilateral with $\angle DAB=60^o$, $\angle ABC=60^o$ and $\angle BCD=120^o$. Diagonals AC,BD intersect at point $M$ and $BM=a, MD=2a$. Let $O$ be the midpoint of side $AC$ and draw $OH \perp BD, H \in BD$ and $MN\perp OB, N \in OB$. Prove that
i) $HM=MN=\frac{a}{2}$
ii) $AD=DC$
iii) $S_{ABCD}=\frac{9a^2}{2}$

2002 Greece JBMO TST P3
Let $ABC$ be a triangle with $\angle A=60^o, AB\ne AC$ and let $AD$ be the angle bisector of $\angle A$. Line $(e)$ that is perpendicular on the angle bisector $AD$ at point $A$, intersects the extension of side $BC$ at point $E$ and also $BE=AB+AC$. Find the angles $\angle B$ and $\angle C$ of the triangle $ABC$.

2003 Greece JBMO TST P4
Given are two points $B,C$. Consider point $A$ not lying on the line $BC$ and draw the circles $C_1(K_1,R_1)$ (with center $K_1$ and radius $R_1$) and $C_2(K_2,R_2)$ with chord $AB, AC$ respectively such that their centers lie on the interior of the triangle $ABC$ and also $R_1 \cdot AC= R_2 \cdot AB$. Let $T$ be the intersection point of the two circles, different from $A$, and M be a random pointof line $AT$, prove that $TC \cdot S_{(MBT)}=TB \cdot S_{(MCT)}$

2004 Greece JBMO TST P1
Let $ABCD$ be a convex quadrilateral with $\angle A=60^o$. Let $E$ and $Z$ be the symmetric points of $A$ wrt $BC$ and $CD$ respectively. If the points $B,D,E$ and $Z$ are collinear, then calculate the angle $\angle BCD$.

2005 Greece JBMO TST P3
Let the midpoint$M$ of the side$AB$ of an inscribed quardiletar, $ABCD$.Let $P$ the point of intersection of $MC$ with $BD$. Let the parallel from the point $C$ to the $AP$ which intersects the $BD$ at$S$. If angle $CAD$ = angle $PAB$ = angle $\frac{BMC}{2}$ , prove that $BP=SD$.

2006 Greece JBMO TST P3
Find the angle $\angle A$ of a triangle $ABC$, when we know it's altitudes $BD$ and $CE$ intersect in an interior point $H$ of the triangle and $BH=2HD$ and $CH=HE$.

Let $ABC$ be a triangle with $\angle A=105^o$ and $\angle C=\frac{1}{4} \angle B$.
a) Find the angles   $\angle B$ and  $\angle C$
b) Let $O$ be the center of the circumscribed circle of the triangle $ABC$ and let $BD$ be a diameter of that circle. Prove that the distance of point $C$ from the line $BD$ is equal to $\frac{BD}{4}$.

2007 Greece JBMO TST P3

2008 Greece JBMO TST P1
Given a point $A$ that lies on circle $c(o,R)$ (with center $O$ and radius $R$). Let $(e)$ be the tangent of the circle $c$ at point $A$ and a line $(d)$ that passes through point $O$ and intersects $(e)$ at point $M$ and the circle at points $B,C$ (let $B$ lie between $O$ and $A$). If $AM = R\sqrt3$ , prove that
a) triangle $AMC$ is isosceles.
b) circumcenter of triangle  $AMC$ lies on circle $c$ .

2009 Greece JBMO TST P2
Given convex quadrilateral $ABCD$ inscribed in circle $(O,R)$ (with center $O$ and radius $R$). With centers the vertices of the quadrilateral and radii $R$, we consider the circles $C_A(A,R), C_B(B,R), C_C(C,R), C_D(D,R)$. Circles $C_A$ and $C_B$ intersect at point $K$, circles $C_B$ and $C_C$ intersect at point $L$, circles $C_C$ and $C_D$ intersect at point $M$ and circles $C_D$ and $C_A$ intersect at point $N$ (points $K,L,M,N$ are the second common points of the circles given they all pass through point $O$). Prove that quadrilateral $KLMN$ is a parallelogram.

2010 Greece JBMO TST P3
Given an acute and scalene triangle $ABC$ with $AB<AC$ and random line $(e)$ that passes throuh the center of the circumscribed circles $c(O,R)$. Line $(e)$, intersects sides $BC,AC,AB$ at points $A_1,B_1,C_1$ respectively (point $C_1$ lies on the extension of $AB$ towards $B$). Perpendicular from $A$ on line $(e)$ and $AA_1$ intersect circumscribed circle $c(O,R)$ at points $M$ and $A_2$ respectively. Prove that
a) points $O,A_1,A_2, M$ are consyclic
b) if $(c_2)$ is the circumcircle of triangle $(OBC_1)$ and $(c_3)$ is the circumcircle of triangle $(OCB_1)$, then circles $(c_1),(c_2)$ and $(c_3)$ have a common chord

2011 Greece JBMO TST P4
Let $ABC$ be an acute and scalene triangle with $AB<AC$, inscribed in a circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ intersects side $BC$ at point $E$ and circle $c$ at point $F$. $EF$ intersects for the second time circle $c$ at point $D$ and side $AC$ at point $M$. $AD$ intersects $BC$ at point $K$. Circumcircle of triangle $BKD$ intersects $AB$ at point $L$ . Prove that points $K,L,M$ lie on a line parallel to $BF$.

2012 Greece JBMO TST P3
Let $ABC$ be an acute triangle with $AB<AC<BC$, inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Let $O_1$ be the symmetric point of $O$ wrt $AC$. Circle $c_1(O_1,R)$ intersects $BC$ at $Z$. If the extension of the altitude $AD$ intersects  the cicrumscribed circle  $c(O,R)$ at point $E$, prove that $EC$ is perpendicular on $AZ$.

2013 Greece JBMO TST P4
Given the circle $c(O,R)$ (with center $O$ and radius $R$), one diameter $AB$ and midpoint $C$ of the arc $AB$. Consider circle $c_1(K,KO)$, where center $K$ lies on the segment $OA$, and consider the tangents $CD,CO$ from the point $C$ to  circle $c_1(K,KO)$. Line $KD$ intersects circle $c(O,R)$ at points $E$ and $Z$ (point $E$ lies on the semicircle that lies also point $C$). Lines $EC$ and $CZ$ intersects $AB$ at points $N$ and $M$ respectively. Prove that quadrilateral $EMZN$ is an isosceles trapezoid, inscribed in a circle whose center lie on circle $c(O,R)$.

2014 Greece JBMO TST P2
Let $ABCD$ be an inscribed quadrilateral in a circle $c(O,R)$ (of circle $O$ and radius $R$). With centers the vertices $A,B,C,D$, we consider the circles  $C_{A},C_{B},C_{C},C_{D}$ respectively, that do not intersect to each other . Circle $C_{A}$ intersects the sides of the quadrilateral at points $A_{1} , A_{2}$ ,  circle $C_{B}$ intersects the sides of the quadrilateral at points $B_{1} , B_{2}$ , circle $C_{C}$ at points $C_{1} , C_{2}$ and circle $C_{D}$ at points $C_{1} , C_{2}$ . Prove that the quadrilateral defined by lines $A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}$ is cyclic.

2015 Greece JBMO TST P2
Let $ABC$ be an acute triangle inscribed in a circle of center $O$. If the altitudes $BD,CE$ intersect at $H$ and the circumcenter of $\triangle BHC$ is $O_1$, prove that $AHO_1O$ is a parallelogram.

2016 Greece JBMO TST P1 (JBMO Shortlist 2015 G3)
Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$.

Evangelos Psychas
2017 Greece JBMO TST P2
Let $ABC$ be an acute-angled triangle inscribed in a circle $\mathcal C (O, R)$ and $F$ a point on the side $AB$ such that $AF < AB/2$. The circle $c_1(F, FA)$ intersects the line $OA$ at the point $A'$ and the circle $\mathcal C$ at $K$. Prove that the quadrilateral $BKFA'$ is cyclic and its circumcircle contains point $O$.

2018 Greece JBMO TST P2
Let $ABC$ be an acute triangle with $AB<AC<BC, c$ it's circumscribed circle and $D,E$ be the midpoints of $AB,AC$ respectively. With diameters the sides $AB,AC$, we draw semicircles, outer of the triangle, which are intersected by line $D$ at points $M$ and $N$ respectively. Lines $MB$ and $NC$ intersect the circumscribed circle at points $T,S$ respectively. Lines $MB$ and $NC$ intersect at point $H$. Prove that:
a) point $H$ lies on the circumcircle of triangle $AMN$
b) lines $AH$ and $TS$ are perpedicular and their intersection, let it be $Z$, is the circimcenter of triangle $AMN$

Consider an acute triangle $ABC$ with $AB>AC$ inscribed in a circle of center $O$. From the midpoint $D$ of side $BC$ we draw line $(\ell)$ perpendicular to side $AB$ that intersects it at point $E$. If line $AO$ intersects line $(\ell)$ at point $Z$, prove that points $A,Z,D,C$ are concyclic.