geometry problems from Greek National Math Olympiads (Seniors)
with aops links in the names
1995 Greek National MO P2
Let $ABC$ be a triangle with $AB = AC$ and let $D$ be a point on $BC$ such that the incircle of $ABD$ and the excircle of $ADC$ corresponding to $A$ have the same radius. Prove that this radius is equal to one quarter of the altitude from $B$ of triangle $ABC$.
1996 Greek National MO P2
Let $ ABC$ be an acute triangle, $ AD,BE,CZ$ its altitudes and $ H$ its orthocenter. Let $ AI,A \Theta$ be the internal and external bisectors of angle $ A$. Let $ M,N$ be the midpoints of $ BC,AH$, respectively. Prove that:
a) $MN$ is perpendicular to $EZ$
b) if $ MN$ cuts the segments $ AI,A \Theta$ at the points $ K,L$, then $ KL= AH$
1997 Greek National MO P1
Let $P$ be a point inside or on the boundary of a square $ABCD$. Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$.
1998 Greek National MO P2
For a regular $n$-gon, let $M$ be the set of the lengths of the segments joining its vertices. Show that the sum of the squares of the elements of $M$ is greater than twice the area of the polygon.
1999 Greek National MO P3
In an acute-angled triangle $ABC$, $AD,BE$ and $CF$ are the altitudes and $H$ the orthocentre. Lines $EF$ and $BC$ meet at $N$. The line passing through $D$ and parallel to $FE$ meets lines $AB$ and $AC$ at $K$ and $L$, respectively. Prove that the circumcircle of the triangle $NKL$ bisects the side $BC$.
2000 Greek National MO P1
Consider a rectangle $ABCD$ with $AB = a$ and $AD = b.$ Let $l$ be a line through $O,$ the center of the rectangle, that cuts $AD$ in $E$ such that $AE/ED = 1/2$. Let $M$ be any point on $l,$ interior to the rectangle. Find the necessary and sufficient condition on $a$ and $b$ that the four distances from M to lines $AD, AB, DC, BC$ in this order form an arithmetic progression.
2001 Greek National MO P1
A triangle $ABC$ is inscribed in a circle of radius $R.$ Let $BD$ and $CE$ be the bisectors of the angles $B$ and $C$ respectively and let the line $DE$ meet the arc $AB$ not containing $C$ at point $K.$ Let $A_1, B_1, C_1$ be the feet of perpendiculars from $K$ to $BC, AC, AB,$ and $x, y$ be the distances from $D$ and $E$ to $BC,$ respectively.
a) Express the lengths of $KA_1, KB_1, KC_1$ in terms of $x, y$ and the ratio $l = KD/ED.$
b) Prove that $\frac{1}{KB}=\frac{1}{KA}+\frac{1}{KC}.$
2002 Greek National MO P3
In a triangle $ABC$ we have $\angle C>10^0$ and $\angle B=\angle C+10^0.$We consider point $E$ on side $AB$ such that $\angle ACE=10^0,$ and point $D$ on side $AC$ such that $\angle DBA=15^0.$ Let $Z\neq A$ be a point of interection of the circumcircles of the triangles $ABD$ and $AEC.$ Prove that $\angle ZBA>\angle ZCA.$
2003 Greek National MO P3
Given are a circle $\mathcal{C}$ with center $K$ and radius $r,$ point $A$ on the circle and point $R$ in its exterior. Consider a variable line $e$ through $R$ that intersects the circle at two points $B$ and $C.$ Let $H$ be the orthocenter of triangle $ABC.$ Show that there is a unique point $T$ in the plane of circle $\mathcal{C}$ such that the sum $HA^2 + HT^2$ remains constant (as $e$ varies.)
2004 Greek National MO P3
Consider a circle $K(O,r)$ and a point $A$ outside $K.$ A line $\epsilon$ different from $AO$ cuts $K$ at $B$ and $C,$ where $B$ lies between $A$ and $C.$ Now the symmetric line of $\epsilon$ with respect to axis of symmetry the line $AO$ cuts $K$ at $E$ and $D,$ where $E$ lies between $A$ and $D.$ Show that the diagonals of the quadrilateral $BCDE$ intersect in a fixed point.
2005 Greek National MO P4
Let $OX_1 , OX_2$ be rays in the interior of a convex angle $XOY$ such that $\angle XOX_1=\angle YOY_1< \frac{1}{3}\angle XOY$. Points $K$ on $OX_1$ and $L$ on $OY_1$ are fixed so that $OK=OL$, and points $A$, $B$ are vary on rays $(OX , (OY$ respectively such that the area of the pentagon $OAKLB$ remains constant. Prove that the circumcircle of the triangle $OAB$ passes from a fixed point, other than $O$.
2006 Greek National MO P3
Let a triangle $ABC$ and the cevians $AL, BN , CM$ such that $AL$ is the bisector of angle $A$. If $\angle ALB = \angle ANM$, prove that $\angle MNL = 90$
2007 Greek National MO P3
In a circular ring with radii $11r$ and $9r$, we put circles of radius $r$ which are tangent to the boundary circles and do not overlap. Determine the maximum number of circles that can be put this way. (You may use that $9.94<\sqrt{99}<9.95$)
2008 Greek National MO P3
A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$, where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK -SH) = 1$, compute the area of the concave quadrilateral $ABHC$.
2009 Greek National MO P2
Consider a triangle $ABC$ with circumcenter $O$ and let $A_1,B_1,C_1$ be the midpoints of the sides $BC,AC,AB,$ respectively. Points $A_2,B_2,C_2$ are defined as $\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1},$ where $\lambda >0.$ Prove that lines $AA_2,BB_2,CC_2$ are concurrent.
2010 Greek National MO P3
A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively. Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic.
2011 Greek National MO P4
We consider an acute angled triangle $ABC$ (with $AB<AC$) and its circumcircle $c(O,R) $(with center $O$ and semidiametre $R$).The altitude $AD$ cuts the circumcircle at the point $E$ ,while the perpedicular bisector $(m)$ of the segment $AB$,cuts $AD$ at the point $L$.$BL$ cuts $AC$ at the point $M$ and the circumcircle $c(O,R)$ at the point $N$.Finally $EN$ cuts the perpedicular bisector $(m)$ at the point $Z$.Prove that:
$ MZ \perp BC \iff \left(CA=CB \;\; \text{or} \;\; Z\equiv O \right) $
2012 Greek National MO P3
Let an acute-angled triangle $ABC$ with $AB<AC<BC$, inscribed in circle $c(O,R)$. The angle bisector $AD$ meets $c(O,R)$ at $K$. The circle $c_1(O_1,R_1)$(which passes from $A,D$ and has its center $O_1$ on $OA$) meets $AB$ at $E$ and $AC$ at $Z$. If $M,N$ are the midpoints of $ZC$ and $BE$ respectively, prove that:
a) the lines $ZE,DM,KC$ are concurrent at one point $T$.
b) the lines $ZE,DN,KB$ are concurrent at one point $X$.
c) $OK$ is the perpendicular bisector of $TX$.
2013 Greek National MO P4
Let a triangle $ABC$ inscribed in circle $c(O,R)$ and $D$ an arbitrary point on $BC$(different from the midpoint).The circumscribed circle of $BOD$,which is $(c_1)$, meets $c(O,R)$ at $K$ and $AB$ at $Z$.The circumscribed circle of $COD$ $(c_2)$,meets $c(O,R)$ at $M$ and $AC$ at $E$.Finally, the circumscribed circle of $AEZ$ $(c_3)$,meets $c(O,R)$ at $N$.Prove that $\triangle{ABC}=\triangle{KMN}.$
2014 Greek National MO P4
We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic
2015 Greek National MO P3
Given is a triangle $ABC$ with $\angle{B}=105^{\circ}$.Let $D$ be a point on $BC$ such that $\angle{BDA}=45^{\circ}$.
a) If $D$ is the midpoint of $BC$ then prove that $\angle{C}=30^{\circ}$,
b) If $\angle{C}=30^{\circ}$ then prove that $D$ is the midpoint of $BC$
2016 Greek National MO P3
$ABC$ is an acute isosceles triangle $(AB=AC)$ and $CD$ one altitude. Circle $C_2(C,CD)$ meets $AC$ at $K$, $AC$ produced at $Z$ and circle $C_1(B, BD)$ at $E$. $DZ$ meets circle $(C_1)$ at $M$. Show that:
a) $\widehat{ZDE}=45^0$
b) Points $E, M, K$ lie on a line.
c) $BM//EC$
2017 Greek National MO P1
An acute triangle $ABC$ with $AB<AC<BC$ is inscribed in a circle $c(O,R)$. The circle $c_1(A,AC)$ intersects the circle $c$ at point $D$ and intersects $CB$ at $E$. If the line $AE$ intersects $c$ at $F$ and $G$ lies in $BC$ such that $EB=BG$, prove that $F,E,D,G$ are concyclic.
2018 Greek National MO P2
Let $ABC$ be an acute-angled triangle with $AB<AC<BC$ and $c(O,R)$ the circumscribed circle. Let $D, E$ be points in the small arcs $AC, AB$ respectively. Let $K$ be the intersection point of $BD,CE$ and $N$ the second common point of the circumscribed circles of the triangles $BKE$ and $CKD$. Prove that $A, K, N$ are collinear if and only if $K$ belongs to the symmedian of $ABC$ passing from $A$
2019 Greek National MO P2
Let $ABC$ be a triangle with $AB<AC<BC$.Let $O$ be the center of it's circumcircle and $D$ be the center of minor arc $AB$.Line $AD$ intersects $BC$ at $E$ and the circumcircle of $BDE$ intersects $AB$ at $Z$ ,($Z\not=B$).The circumcircle of $ADZ$ intersects $AC$ at $H$ ,($H\not=A$),prove that $BE=AH$
2020 Greek National MO P2
Given a line segment $AB$ and a point $C$ lies inside it such that $AB=3 \cdot AC$ . Construct a parallelogram $ACDE$ such that $AC=DE=CE>AR$. Let $Z$ be a point on $AC$ such that $\angle AEZ=\angle ACE =\omega$. Prove that the line passing through point $B$ and perpendicular on side $EC$, and the line passing through point $D$ and perpendicular on side $AB$, intersect on point , let it be $K$, lying on line $EZ$.
with aops links in the names
1995 - 2020, 2022
Let $ABC$ be a triangle with $AB = AC$ and let $D$ be a point on $BC$ such that the incircle of $ABD$ and the excircle of $ADC$ corresponding to $A$ have the same radius. Prove that this radius is equal to one quarter of the altitude from $B$ of triangle $ABC$.
1996 Greek National MO P2
Let $ ABC$ be an acute triangle, $ AD,BE,CZ$ its altitudes and $ H$ its orthocenter. Let $ AI,A \Theta$ be the internal and external bisectors of angle $ A$. Let $ M,N$ be the midpoints of $ BC,AH$, respectively. Prove that:
a) $MN$ is perpendicular to $EZ$
b) if $ MN$ cuts the segments $ AI,A \Theta$ at the points $ K,L$, then $ KL= AH$
1997 Greek National MO P1
Let $P$ be a point inside or on the boundary of a square $ABCD$. Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$.
1998 Greek National MO P2
For a regular $n$-gon, let $M$ be the set of the lengths of the segments joining its vertices. Show that the sum of the squares of the elements of $M$ is greater than twice the area of the polygon.
1999 Greek National MO P3
In an acute-angled triangle $ABC$, $AD,BE$ and $CF$ are the altitudes and $H$ the orthocentre. Lines $EF$ and $BC$ meet at $N$. The line passing through $D$ and parallel to $FE$ meets lines $AB$ and $AC$ at $K$ and $L$, respectively. Prove that the circumcircle of the triangle $NKL$ bisects the side $BC$.
2000 Greek National MO P1
Consider a rectangle $ABCD$ with $AB = a$ and $AD = b.$ Let $l$ be a line through $O,$ the center of the rectangle, that cuts $AD$ in $E$ such that $AE/ED = 1/2$. Let $M$ be any point on $l,$ interior to the rectangle. Find the necessary and sufficient condition on $a$ and $b$ that the four distances from M to lines $AD, AB, DC, BC$ in this order form an arithmetic progression.
2001 Greek National MO P1
A triangle $ABC$ is inscribed in a circle of radius $R.$ Let $BD$ and $CE$ be the bisectors of the angles $B$ and $C$ respectively and let the line $DE$ meet the arc $AB$ not containing $C$ at point $K.$ Let $A_1, B_1, C_1$ be the feet of perpendiculars from $K$ to $BC, AC, AB,$ and $x, y$ be the distances from $D$ and $E$ to $BC,$ respectively.
a) Express the lengths of $KA_1, KB_1, KC_1$ in terms of $x, y$ and the ratio $l = KD/ED.$
b) Prove that $\frac{1}{KB}=\frac{1}{KA}+\frac{1}{KC}.$
2002 Greek National MO P3
In a triangle $ABC$ we have $\angle C>10^0$ and $\angle B=\angle C+10^0.$We consider point $E$ on side $AB$ such that $\angle ACE=10^0,$ and point $D$ on side $AC$ such that $\angle DBA=15^0.$ Let $Z\neq A$ be a point of interection of the circumcircles of the triangles $ABD$ and $AEC.$ Prove that $\angle ZBA>\angle ZCA.$
2003 Greek National MO P3
Given are a circle $\mathcal{C}$ with center $K$ and radius $r,$ point $A$ on the circle and point $R$ in its exterior. Consider a variable line $e$ through $R$ that intersects the circle at two points $B$ and $C.$ Let $H$ be the orthocenter of triangle $ABC.$ Show that there is a unique point $T$ in the plane of circle $\mathcal{C}$ such that the sum $HA^2 + HT^2$ remains constant (as $e$ varies.)
2004 Greek National MO P3
Consider a circle $K(O,r)$ and a point $A$ outside $K.$ A line $\epsilon$ different from $AO$ cuts $K$ at $B$ and $C,$ where $B$ lies between $A$ and $C.$ Now the symmetric line of $\epsilon$ with respect to axis of symmetry the line $AO$ cuts $K$ at $E$ and $D,$ where $E$ lies between $A$ and $D.$ Show that the diagonals of the quadrilateral $BCDE$ intersect in a fixed point.
2005 Greek National MO P4
Let $OX_1 , OX_2$ be rays in the interior of a convex angle $XOY$ such that $\angle XOX_1=\angle YOY_1< \frac{1}{3}\angle XOY$. Points $K$ on $OX_1$ and $L$ on $OY_1$ are fixed so that $OK=OL$, and points $A$, $B$ are vary on rays $(OX , (OY$ respectively such that the area of the pentagon $OAKLB$ remains constant. Prove that the circumcircle of the triangle $OAB$ passes from a fixed point, other than $O$.
2006 Greek National MO P3
Let a triangle $ABC$ and the cevians $AL, BN , CM$ such that $AL$ is the bisector of angle $A$. If $\angle ALB = \angle ANM$, prove that $\angle MNL = 90$
2007 Greek National MO P3
In a circular ring with radii $11r$ and $9r$, we put circles of radius $r$ which are tangent to the boundary circles and do not overlap. Determine the maximum number of circles that can be put this way. (You may use that $9.94<\sqrt{99}<9.95$)
2008 Greek National MO P3
A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$, where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK -SH) = 1$, compute the area of the concave quadrilateral $ABHC$.
2009 Greek National MO P2
Consider a triangle $ABC$ with circumcenter $O$ and let $A_1,B_1,C_1$ be the midpoints of the sides $BC,AC,AB,$ respectively. Points $A_2,B_2,C_2$ are defined as $\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1},$ where $\lambda >0.$ Prove that lines $AA_2,BB_2,CC_2$ are concurrent.
2010 Greek National MO P3
A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively. Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic.
2011 Greek National MO P4
We consider an acute angled triangle $ABC$ (with $AB<AC$) and its circumcircle $c(O,R) $(with center $O$ and semidiametre $R$).The altitude $AD$ cuts the circumcircle at the point $E$ ,while the perpedicular bisector $(m)$ of the segment $AB$,cuts $AD$ at the point $L$.$BL$ cuts $AC$ at the point $M$ and the circumcircle $c(O,R)$ at the point $N$.Finally $EN$ cuts the perpedicular bisector $(m)$ at the point $Z$.Prove that:
$ MZ \perp BC \iff \left(CA=CB \;\; \text{or} \;\; Z\equiv O \right) $
2012 Greek National MO P3
Let an acute-angled triangle $ABC$ with $AB<AC<BC$, inscribed in circle $c(O,R)$. The angle bisector $AD$ meets $c(O,R)$ at $K$. The circle $c_1(O_1,R_1)$(which passes from $A,D$ and has its center $O_1$ on $OA$) meets $AB$ at $E$ and $AC$ at $Z$. If $M,N$ are the midpoints of $ZC$ and $BE$ respectively, prove that:
a) the lines $ZE,DM,KC$ are concurrent at one point $T$.
b) the lines $ZE,DN,KB$ are concurrent at one point $X$.
c) $OK$ is the perpendicular bisector of $TX$.
2013 Greek National MO P4
Let a triangle $ABC$ inscribed in circle $c(O,R)$ and $D$ an arbitrary point on $BC$(different from the midpoint).The circumscribed circle of $BOD$,which is $(c_1)$, meets $c(O,R)$ at $K$ and $AB$ at $Z$.The circumscribed circle of $COD$ $(c_2)$,meets $c(O,R)$ at $M$ and $AC$ at $E$.Finally, the circumscribed circle of $AEZ$ $(c_3)$,meets $c(O,R)$ at $N$.Prove that $\triangle{ABC}=\triangle{KMN}.$
2014 Greek National MO P4
We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic
2015 Greek National MO P3
Given is a triangle $ABC$ with $\angle{B}=105^{\circ}$.Let $D$ be a point on $BC$ such that $\angle{BDA}=45^{\circ}$.
a) If $D$ is the midpoint of $BC$ then prove that $\angle{C}=30^{\circ}$,
b) If $\angle{C}=30^{\circ}$ then prove that $D$ is the midpoint of $BC$
2016 Greek National MO P3
$ABC$ is an acute isosceles triangle $(AB=AC)$ and $CD$ one altitude. Circle $C_2(C,CD)$ meets $AC$ at $K$, $AC$ produced at $Z$ and circle $C_1(B, BD)$ at $E$. $DZ$ meets circle $(C_1)$ at $M$. Show that:
a) $\widehat{ZDE}=45^0$
b) Points $E, M, K$ lie on a line.
c) $BM//EC$
2017 Greek National MO P1
An acute triangle $ABC$ with $AB<AC<BC$ is inscribed in a circle $c(O,R)$. The circle $c_1(A,AC)$ intersects the circle $c$ at point $D$ and intersects $CB$ at $E$. If the line $AE$ intersects $c$ at $F$ and $G$ lies in $BC$ such that $EB=BG$, prove that $F,E,D,G$ are concyclic.
2018 Greek National MO P2
Let $ABC$ be an acute-angled triangle with $AB<AC<BC$ and $c(O,R)$ the circumscribed circle. Let $D, E$ be points in the small arcs $AC, AB$ respectively. Let $K$ be the intersection point of $BD,CE$ and $N$ the second common point of the circumscribed circles of the triangles $BKE$ and $CKD$. Prove that $A, K, N$ are collinear if and only if $K$ belongs to the symmedian of $ABC$ passing from $A$
2019 Greek National MO P2
Let $ABC$ be a triangle with $AB<AC<BC$.Let $O$ be the center of it's circumcircle and $D$ be the center of minor arc $AB$.Line $AD$ intersects $BC$ at $E$ and the circumcircle of $BDE$ intersects $AB$ at $Z$ ,($Z\not=B$).The circumcircle of $ADZ$ intersects $AC$ at $H$ ,($H\not=A$),prove that $BE=AH$
2020 Greek National MO P2
Given a line segment $AB$ and a point $C$ lies inside it such that $AB=3 \cdot AC$ . Construct a parallelogram $ACDE$ such that $AC=DE=CE>AR$. Let $Z$ be a point on $AC$ such that $\angle AEZ=\angle ACE =\omega$. Prove that the line passing through point $B$ and perpendicular on side $EC$, and the line passing through point $D$ and perpendicular on side $AB$, intersect on point , let it be $K$, lying on line $EZ$.
Let $ABC$ be a triangle such that $AB<AC<BC$. Let $D,E$ be points on the segment $BC$ such that $BD=BA$ and $CE=CA$. If $K$ is the circumcenter of triangle $ADE$, $F$ is the intersection of lines $AD,KC$ and $G$ is the intersection of lines $AE,KB$, then prove that the circumcircle of triangle $KDE$ (let it be $c_1$), the circle with center the point $F$ and radius $FE$ (let it be $c_2$) and the circle with center $G$ and radius $GD$ (let it be $c_3$) concur on a point which lies on the line $AK$.
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