### Greece Junior 1998 - 2019 23p

geometry problems from Greek Junior Mathematical Olympiads (JMO) [Archimedes Junior]

1998 - 2019

1998 Greece JMO P4
Let $K(O,R)$ be a circle with center $O$ and radious $R$ and $(e)$ to be a line thst tangent to $K$ at $A$. A line parallel to $OA$ cuts $K$ at $B, C$, and $(e)$ at $D$, ($C$ is between $B$ and $D$). Let $E$ to be the antidiameric of $C$ with respect to $K$. $EA$ cuts $BD$ at $F$.
i) Examine if $CEF$ is isosceles.
ii) Prove that $2AD=EB$.
iii) If $K$ is the midpoint of $CF$, prove that $AB=KO$.
iv) If $R=\frac{5}{2}, AD=\frac{3}{2}$, calculate the area of $EBF$

1999 Greece JMO P3
Let $ABC$ be an equilateral triangle . Let  point $D$ lie on side $AB,E$  lie on side $AC, D_1$ and $E_1$ lie on side BC such that $AB=DB+BD_1$ and $AC=CE+CE_1$. Calculate the smallest angle between the lines $DE_1$ and $ED_1$.

2000 Greece JMO P1
Given are three non collinear points in plane. Find a line from which all three points are equidistant. How many such lines of the plane exist?

2001 Greece JMO P4
Let $ABC$ be a triangle with altitude $AD$ , angle bisectors $AE$ and $BZ$ that intersecting at point $I$. From point $I$ let $IT$ be a perpendicular on $AC$. Also let line $(e)$ be perpendicular on $AC$ at point $A$. Extension of $ET$ intersects line $(e)$ at point K. Prove that $AK=AD$.

2002 Greece JMO P1
Οn the exterior of an equilateral triangle $ABC$ of side $a$ we construct a right isosceles triangle $ACD$ with $\angle CAD=90^o$. The extensions of segments $DA$ and $CB$ intersect at point $E$.
i) Calculate the measure of the angle $\angle DBC$.
ii) Calculate the area of the triangle $CDE$ in terms of $a$
iii)  Calculate the length of $BD$ in terms of $a$.

2003 Greece JMO P3
Let $ABC$ be an isosceles triangle  ($AB=AC$). The altitude $AH$ and the perpendiculare bisector $(e)$ of side $AB$ intersect at point $M$ . The perpendicular on line $(e)$ passing through $M$ intersects $BC$ at point $D$. If the circumscribed circle of the triangle $BMD$  intersects line $(e)$ at point $S$ , the prove that:
a) $BS // AM$ .
b) quadrilateral $AMBS$ is rhombus.

2004 Greece JMO P2
Let $ABCD$ be a rectangle. Let $K,L$ be the midpoints of $BC, AD$ respectively. From point $B$ the perpendicular line on $AK$, intersects $AK$ at point $E$ and $CL$ at point $Z$.
a) Prove that the quadrilateral $AKZL$ is an isosceles trapezoid
b) Prove that $2S_{ABKZ}=S_{ABCD}$
c) If quadrilateral $ABCD$ is a square of side $a$, calculate the area of the  isosceles trapezoid $AKZL$ in terms of side $BC=a$

2005 Greece JMO P1
Let $ABCD$ be a trapezoid with $AB//CD, AB=a, CD=2a$ and $DB \perp BC$. Let $M$ be the mdipoint of $CD, O$ the intersection point of the diagonals of $ABMD, K$ the intersection point of lines $DA,CB$ and $L$ be the intersection point of lines $KO$ and $AB$. Prove that
i) quadrilateral $ABMD$ is rhombus
ii) triangles $CDK$ is isosceles
iii) lines $DL$ intersects segment $KB$ at it's midpoint.

2005 Greece JMO P4
Let $\omega$ be a circle of center $O$ and radius $R$, and let $A$ be an exterior point with $AO=d$. Find points $B,C,D$ that lie on circle $\omega$, such that a convex quadrilateral $ABCD$ can be constructed with greatest possible area.

2006 Greece JMO P1
Let $P$  an interior point of an equilateral triangle  $ABC$. Prove that there exists triangle with sides $PA , PB , PC$ .

2007 Greece JMO P1
Let $ABC$ be a triagle with $AB<AC$. Let $I$ be the intersection point of it's angle bisectors. Angle bisector $AD$ intersects the circumscribed circle $\omega$ of the triangle $BIC$ at point $N$ with $N \ne I$.
a) Calculate the angles of triangle $BCN$ in terms of the angles of triangle $ABC$.
b) Find the center of the circle  $\omega$

2008 Greece JMO P4
Let $ABCD$ be a trapezoid with $AD=a, AB=2a, BC=3a$ and $\angle A=\angle B =90 ^o$. Let $E,Z$ be the midpoints of the sides $AB ,CD$ respectively and $I$ be the foot of the perpendicular from point $Z$ on $BC$. Prove that :
i) triangle $BDZ$ is isosceles
ii) midpoint $O$ of $EZ$ is the centroid of triangle $BDZ$
iii) lines $AZ$ and $DI$ intersect at a point lying on line $BO$

2009 Greece JMO P2
From vertex $A$ of an equilateral triangle $ABC$, a ray $Ax$ intersects $BC$ at point $D$. Let $E$ be a point on $Ax$ such that $BA =BE$. Calculate $\angle AEC$.

2010 Greece JMO P2
Let $ABCD$ be a rectangle with sides $AB=a$ and $BC=b$. Let $O$ be the intersection point of it's diagonals. Extent side $BA$ towards $A$ at a segment $AE=AO$, and diagonal $DB$ towards $B$ at a segment $BZ=BO$. If the triangle $EZC$ is an equilateral, then prove that:
i) $b=a\sqrt3$
ii) $AZ=EO$
iii) $EO \perp ZD$

2011 Greece JMO P1
Let $ABC$ be a triangle with $\angle BAC=120^o$, which the median $AD$ is perpendicular to side $AB$ and intersects the circumscribed circle of triangle $ABC$ at point $E$. Lines $BA$ and $EC$ intersect at $Z$. Prove that
a) $ZD \perp BE$
b) $ZD=BC$

2012 Greece JMO P1
Let $ABC$ be an acute angled triangle (with $AB<AC<BC$) inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ (with center $A$ and radius $AB$) intersects side $BC$ at point $D$ and the circumcircle $c(O,R)$ at point $E$. Prove that side $AC$ bisects angle $\angle DAE$.

2013 Greece JMO P2
Let $ABC$ be an acute angled triangle with $AB<AC$. Let $M$ be the midpoint of side $BC$. On side $AB$, consider a point $D$ such that, if segment $CD$ intersects median $AM$ at point $E$, then $AD=DE$. Prove that $AB=CE$.

2014 Greece JMO P1
Let $ABC$ be a triangle and let $M$ be the midpoint $BC$. On the exterior of the triangle, consider the parallelogram $BCDE$ such that $BE//AM$ and $BE=AM/2$ . Prove that line $AM$ passes through the midpoint of segment $AD$.

2015 Greece JMO P4
Let $ABC$ be an acute triangle with $AB\le AC$ and let $c(O,R)$ be it's circumscribed circle (with center $O$ and radius $R$). The perpendicular from vertex $A$ on the tangent of the circle passing through point $C$, intersect it at point $D$.
a) If the triangle $ABC$ is isosceles with $AB=AC$, prove that $CD=BC/2$.
b) If $CD=BC/2$, prove that the triangle $ABC$ is isosceles.

2016 Greece JMO P3
Let $ABCD$ be a trapezoid ($AD//BC$) with $\angle A=\angle B= 90^o$ and $AD<BC$. Let $E$ be the intersection point of the non parallel sides $AB$ and $CD$,  $Z$ be the symmetric point of $A$ wrt line $BC$ and $M$ be the midpoint of $EZ$. If it is given than line $CM$ is perpendicular on line $DZ$, then prove that line $ZC$ is perpendicular on line $EC$.

2017 Greece JMO P1
Let $ABCD$ be a square of side $a$. On side $AD$ consider points $E$ and $Z$ such that $DE=a/3$ and $AZ=a/4$. If the lines $BZ$ and $CE$ intersect at point $H$, calculate the area of the triangle $BCH$ in terms of $a$.

2018 Greece JMO P4
Let $ABC$ with $AB<AC<BC$ be an acute angled triangle and $c$ its circumcircle. Let $D$ be the point diametrically opposite to $A$. Point $K$ is on $BD$ such that $KB=KC$. The circle $(K, KC)$ intersects $AC$ at point $E$. Prove that the circle $(BKE)$ is tangent to $c$.

2019 Greece JMO P2
Let $ABCD$ be a quadrilateral inscribed in circle of center $O$. The perpendicular on the midpoint of side $BC$ intersects line $AB$ at point $Z$. The circumscribed circle of the triangle $CEZ$, intersects the side $AB$ for the second time at point $H$ and line $CD$ at point $G$ different than $D$. Line $EG$ intersects line $AD$ at point $K$ and line $CH$ at point $L$. Prove that the points $A,H,L,K$ are concyclic, e.g. lie on the same circle.