geometry problems from Greek Team Selection Tests (TST)
with aops links in the names
2008 Greek TST P3
The bisectors of the angles $\angle{A},\angle{B},\angle{C}$ of a triangle $\triangle{ABC}$ intersect with the circumcircle $c_1(O,R)$ of $\triangle{ABC}$ at $A_2,B_2,C_2$ respectively.The tangents of $c_1$ at $A_2,B_2,C_2$ intersect each other at $A_3,B_3,C_3$ (the points $A_3,A$ lie on the same side of $BC$,the points $B_3,B$ on the same side of $CA$,and $C_3,C$ on the same side of $AB$).The incircle $c_2(I,r)$ of $\triangle{ABC}$ is tangent to $BC,CA,AB$ at $A_1,B_1,C_1$ respectively.Prove that $A_1A_2,B_1B_2,C_1C_2,AA_3,BB_3,CC_3$ are concurrent.
2009 Greek TST P2
Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.
2011 Greek TST P4
Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.
2012 Greek TST P2
Given is an acute triangle $ABC$ $\left(AB<AC<BC\right)$,inscribed in circle $c(O,R)$.The perpendicular bisector of the angle bisector $AD$ $\left(D\in BC\right)$ intersects $c$ at $K,L$ ($K$ lies on the small arc $AB$).The circle $c_1(K,KA)$ intersects $c$ at $T$ and the circle $c_2(L,LA)$ intersects $c$ at $S$.Prove that $\angle{BAT}=\angle{CAS}$.
2013 Greek TST1 P2
Let $ABC$ be a non-isosceles,aqute triangle with $AB<AC$ inscribed in circle $c(O,R)$.The circle $c_{1}(B,AB)$ crosses $AC$ at $K$ and $c$ at $E$.
$KE$ crosses $c$ at $F$ and $BO$ crosses $KE$ at $L$ and $AC$ at $M$ while $AE$ crosses $BF$ at $D$.Prove that:
i) $D,L,M,F$ are concyclic.
ii) $B,D,K,M,E$ are concyclic.
2013 Greek TST2 P3
Given is a triangle $ABC$.On the extensions of the side $AB$ we consider points $A_1,B_1$ such that $AB_1=BA_1$ (with $A_1$ lying closer to $B$).On the extensions of the side $BC$ we consider points $B_4,C_4$ such that $CB_4=BC_4$ (with $B_4$ lying closer to $C$).On the extensions of the side $AC$ we consider points $C_1,A_4$ such that $AC_1=CA_4$ (with $C_1$ lying closer to $A$).On the segment $A_1A_4$ we consider points $A_2,A_3$ such that $A_1A_2=A_3A_4=mA_1A_4$ where $0<m<\frac{1}{2}$.Points $B_2,B_3$ and $C_2,C_3$ are defined similarly,on the segments $B_1B_4,C_1C_4$ respectively.If $D\equiv BB_2\cap CC_2 \ , \ E\equiv AA_3\cap CC_2 \ , \ F\equiv AA_3\cap BB_3$, $\ G\equiv BB_3\cap CC_3 \ , \ H\equiv AA_2\cap CC_3$ and $I\equiv AA_2\cap BB_2$,prove that the diagonals $DG,EH,FI$ of the hexagon $DEFGHI$ are concurrent.
2014 Greek TST P3
Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.
2015 Greek TST P3
Let $ABC$ be an acute triangle with $\displaystyle{AB<AC<BC}$ inscribed in circle $ \displaystyle{c(O,R)}$.The excircle $\displaystyle{(c_A)}$ has center $\displaystyle{I}$ and touches the sides $\displaystyle{BC,AC,AB}$ of the triangle $ABC$ at $\displaystyle{D,E,Z} $ respectively.$ \displaystyle{AI}$ cuts $\displaystyle{(c)}$ at point $M$ and the circumcircle $\displaystyle{(c_1)}$ of triangle $\displaystyle{AZE}$ cuts $\displaystyle{(c)}$ at $K$.The circumcircle $\displaystyle{(c_2)}$ of the triangle $\displaystyle{OKM}$ cuts $\displaystyle{(c_1)} $ at point $N$.Prove that the point of intersection of the lines $AN,KI$ lies on $ \displaystyle{(c)}$.
2016 Greek TST P2
Given is a triangle $\triangle{ABC}$,with $AB<AC<BC$,inscribed in circle $c(O,R)$.Let $D,E,Z$ be the midpoints of $BC,CA,AB$ respectively,and $K$ the foot of the altitude from $A$.At the exterior of $\triangle{ABC}$ and with the sides $AB,AC$ as diameters,we construct the semicircles $c_1,c_2$ respectively.Suppose that $P\equiv DZ\cap c_1 \ , \ S\equiv KZ\cap c_1$ and $R\equiv DE\cap c_2 \ , \ T\equiv KE\cap c_2$.Finally,let $M$ be the intersection of the lines $PS,RT$.
i) Prove that the lines $PR,ST$ intersect at $A$.
ii) Prove that the lines $PR\cap MD$ intersect on $c$.
2017 Greek TST P1
Let $ABC$ be an acute-angled triangle inscribed in circle $c(O,R)$ with $AB<AC<BC$,
and $c_1$ be the inscribed circle of $ABC$ which intersects $AB, AC, BC$ at
$F, E, D$ respectivelly. Let $A', B', C'$ be points which lie on $c$ such that the quadrilaterals
$AEFA', BDFB', CDEC'$ are inscribable.
(1) Prove that $DEA'B'$ is inscridable.
(2) Prove that $DA', EB', FC'$ are concurrent.
2018 Greek TST P2
A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ . We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.
2019 Greek TST P2
Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$. ( $AN$ is a diametre of the circle $\Gamma$)
with aops links in the names
(only those not in IMO Shortlist)
[4 p per day]
2008 - 2020, 2022
The bisectors of the angles $\angle{A},\angle{B},\angle{C}$ of a triangle $\triangle{ABC}$ intersect with the circumcircle $c_1(O,R)$ of $\triangle{ABC}$ at $A_2,B_2,C_2$ respectively.The tangents of $c_1$ at $A_2,B_2,C_2$ intersect each other at $A_3,B_3,C_3$ (the points $A_3,A$ lie on the same side of $BC$,the points $B_3,B$ on the same side of $CA$,and $C_3,C$ on the same side of $AB$).The incircle $c_2(I,r)$ of $\triangle{ABC}$ is tangent to $BC,CA,AB$ at $A_1,B_1,C_1$ respectively.Prove that $A_1A_2,B_1B_2,C_1C_2,AA_3,BB_3,CC_3$ are concurrent.
2009 Greek TST P2
Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.
Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$.
i) Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point.
ii) Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.
Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.
2012 Greek TST P2
Given is an acute triangle $ABC$ $\left(AB<AC<BC\right)$,inscribed in circle $c(O,R)$.The perpendicular bisector of the angle bisector $AD$ $\left(D\in BC\right)$ intersects $c$ at $K,L$ ($K$ lies on the small arc $AB$).The circle $c_1(K,KA)$ intersects $c$ at $T$ and the circle $c_2(L,LA)$ intersects $c$ at $S$.Prove that $\angle{BAT}=\angle{CAS}$.
2013 Greek TST1 P2
Let $ABC$ be a non-isosceles,aqute triangle with $AB<AC$ inscribed in circle $c(O,R)$.The circle $c_{1}(B,AB)$ crosses $AC$ at $K$ and $c$ at $E$.
$KE$ crosses $c$ at $F$ and $BO$ crosses $KE$ at $L$ and $AC$ at $M$ while $AE$ crosses $BF$ at $D$.Prove that:
i) $D,L,M,F$ are concyclic.
ii) $B,D,K,M,E$ are concyclic.
2013 Greek TST2 P3
Given is a triangle $ABC$.On the extensions of the side $AB$ we consider points $A_1,B_1$ such that $AB_1=BA_1$ (with $A_1$ lying closer to $B$).On the extensions of the side $BC$ we consider points $B_4,C_4$ such that $CB_4=BC_4$ (with $B_4$ lying closer to $C$).On the extensions of the side $AC$ we consider points $C_1,A_4$ such that $AC_1=CA_4$ (with $C_1$ lying closer to $A$).On the segment $A_1A_4$ we consider points $A_2,A_3$ such that $A_1A_2=A_3A_4=mA_1A_4$ where $0<m<\frac{1}{2}$.Points $B_2,B_3$ and $C_2,C_3$ are defined similarly,on the segments $B_1B_4,C_1C_4$ respectively.If $D\equiv BB_2\cap CC_2 \ , \ E\equiv AA_3\cap CC_2 \ , \ F\equiv AA_3\cap BB_3$, $\ G\equiv BB_3\cap CC_3 \ , \ H\equiv AA_2\cap CC_3$ and $I\equiv AA_2\cap BB_2$,prove that the diagonals $DG,EH,FI$ of the hexagon $DEFGHI$ are concurrent.
2014 Greek TST P3
Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.
Let $ABC$ be an acute triangle with $\displaystyle{AB<AC<BC}$ inscribed in circle $ \displaystyle{c(O,R)}$.The excircle $\displaystyle{(c_A)}$ has center $\displaystyle{I}$ and touches the sides $\displaystyle{BC,AC,AB}$ of the triangle $ABC$ at $\displaystyle{D,E,Z} $ respectively.$ \displaystyle{AI}$ cuts $\displaystyle{(c)}$ at point $M$ and the circumcircle $\displaystyle{(c_1)}$ of triangle $\displaystyle{AZE}$ cuts $\displaystyle{(c)}$ at $K$.The circumcircle $\displaystyle{(c_2)}$ of the triangle $\displaystyle{OKM}$ cuts $\displaystyle{(c_1)} $ at point $N$.Prove that the point of intersection of the lines $AN,KI$ lies on $ \displaystyle{(c)}$.
2016 Greek TST P2
Given is a triangle $\triangle{ABC}$,with $AB<AC<BC$,inscribed in circle $c(O,R)$.Let $D,E,Z$ be the midpoints of $BC,CA,AB$ respectively,and $K$ the foot of the altitude from $A$.At the exterior of $\triangle{ABC}$ and with the sides $AB,AC$ as diameters,we construct the semicircles $c_1,c_2$ respectively.Suppose that $P\equiv DZ\cap c_1 \ , \ S\equiv KZ\cap c_1$ and $R\equiv DE\cap c_2 \ , \ T\equiv KE\cap c_2$.Finally,let $M$ be the intersection of the lines $PS,RT$.
i) Prove that the lines $PR,ST$ intersect at $A$.
ii) Prove that the lines $PR\cap MD$ intersect on $c$.
2017 Greek TST P1
Let $ABC$ be an acute-angled triangle inscribed in circle $c(O,R)$ with $AB<AC<BC$,
and $c_1$ be the inscribed circle of $ABC$ which intersects $AB, AC, BC$ at
$F, E, D$ respectivelly. Let $A', B', C'$ be points which lie on $c$ such that the quadrilaterals
$AEFA', BDFB', CDEC'$ are inscribable.
(1) Prove that $DEA'B'$ is inscridable.
(2) Prove that $DA', EB', FC'$ are concurrent.
2018 Greek TST P2
A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ . We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.
2019 Greek TST P2
Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$. ( $AN$ is a diametre of the circle $\Gamma$)
Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $M$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.
Consider triangle $ABC$ with $AB<AC<BC$, inscribed in triangle $\Gamma_1$ and the circles $\Gamma_2 (B,AC)$ and $\Gamma_2 (C,AB)$. A common point of circle $\Gamma_2$ and $\Gamma_3$ is point $E$, a common point of circle $\Gamma_1$ and $\Gamma_3$ is point $F$ and a common point of circle $\Gamma_1$ and $\Gamma_2$ is point $G$, where the points $E,F,G$ lie on the same semiplane defined by line $BC$, that point $A$ doesn't lie in. Prove that circumcenter of triangle $EFG$ lies on circle $\Gamma_1$.
Note: By notation $\Gamma (K,R)$, we mean random circle $\Gamma$ has center $K$ and radius $R$.
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