Cyprus TST 2005,2009-21 35p (-11)

  geometry problems from Cyprus Team Selection Tests (TST) with aops links in the names

(only those not in IMO Shortlist)

collected inside aops here

2005, 2009 - 2021 

 2011 missing

2005 Cyprus TST 1.3

Given a circle with ceneter $O$ and an inscribed trapezium $AB\Gamma\Delta$ $(AB\parallel\Gamma\Delta)$ with $AB<\Gamma\Delta$. If $P$ is a point of arc $\Gamma\Delta$ on which $A$ and $B$ do not belong to, and $P_{1},P_{2},P_{3}$ and $P_{4}$ are the projections of $R$ on the lines $A\Delta$, $B\Gamma$, $B\Delta$, and $A\Gamma$ respectively, show that:

(i) The circumscribed circles of the triangles $PP_{1}\Delta$ and $PP_{2}\Gamma$ intersect on the side $\Gamma\Delta$.

(ii) The points $P_{1},P_{2},P_{3}$ and $P_{4}$ are concyclic.

(iii) If $AB=\alpha$, $\Gamma\Delta=\beta$ and the distance between the parallel chords is $h$ find all the points of the axis of symmetry of the trapezium $AB\Gamma\Delta$ that can "see"* at right angle the non parallel sides and calculate their distance form $AB$ and $\Gamma\Delta$ in terms of $\alpha$, $\beta$ and $h$.

Examinate if  such points always exist. 

(Draw a separate diagram for part (iii)).

2005 Cyprus TST 2.3

Given an acute triangle $ABC$ with $AB<AC$. Let $H$ be the orthocenter and $O$ be the circumcenter. The perpendicular bisector of $AH$ intersects $AB$ at $D$ and $AC$ ta $E$. The perpendicular from $O$ on $OD$ intersects the altitudes form vertices $A$ and $C$ at points $T$ and $P$ respectovely. Prove that $TH=TP$.

2009 Cyprus TST 1.4

Given a parallelogram $ABCD$ with $\angle A=60^o$ and let $O$ be the circumcenter of triangle $ABD$. Lines $AO$ and $AB$ intersect the angle bisector of the external angle $\angle C$ of the parallelogram at points K and S respectively. Prove for areas that $(KOS)=2(AOS)$ .

2009 Cyprus TST 2.2

Two unequal circles intersect at the points $M$ and $N$. From a point $D$ of the line $MN$, which is located towards $N$, we draw tangents to the two circles with touching points $S$ and $T$ . The perpendicular lines on the tangents at the points of touching intersect at $K$. Prove that $KM \perp MN$ if and only if the points $S,N,T$ are collinear. 

2010 Cyprus TST 1.2

Given a parallelogram $ABCD$ with $AC>BD$  and $O$ the intersection point of the diagonals. Circle with center $O$ and radius $OA$ intersects the extensions of $AB$ and $AD$ at points $L$ and $K$ respectively. Let $Z$ be the intersection popint of $KL$ and $BD$. If the line passing through $Z$ parallel to $AC$, intersects the tangent of the circle drawn at point $A$, at point $M,$ prove that the quadrilateral $ACZM$ is a rectangle.

2010 Cyprus TST 2.3 [typo]

Let $M$ be the intersection point of the diagonals of a convex quasrliateral $ABCD$, The angle bisector of $\angle ACD$ intersects line $BA$ at point $K$ and the circumscribed circle of the triangle $ABC$ at point $N$. If $T$ is the intersection point with the circle and $MA \cdot MC + MA \cdot CD= MB \cdot MD$ prove that $NA=NT$.

PS: Not well defined point $T$. 

2011 missing

2012 Cyprus TST 1.4

Let $O$ be the center of the circumcircle $(c)$ of the triangle $ABC$ and $AD$ its diamater. Tangent $(e)$ of $(c)$ at point $D$ intersects line $BC$ at point $P$. If $PO$ intersects lines $AC$ and $AB$ at points $M$ and $N$ respectively and the perpendicular on $PO$ at point $O$ intersects $(e)$ at point $E$, prove that $EN=EM$.

2012 Cyprus TST 2.3

Given a tirangle $ABC$ and $Z$ a point interior such that $CZ$ intersects $AB$ at $D, BZ$ intersects $AC$ at $H$ , $AD=BD=CZ$ and $CH=ZH$. If the bisector of angle $\angle BDC$ and the perpendicular bisector of $BC$ intersect at point $E$, prove that tirangle $BCE$ is equilateral.

2012 Cyprus TST 4.2

Given a right trapezoid $ABCD$ with $AB \parallel CD$ and $\angle <A=\angle D=90^o$ and an interior point $E$. From $E$ draw a perpendicular on $AD$ and let $Z$ be it's intersection point with $AD$. IF $K,L,N$ are the feet of the perpendicular from $A,D,Z$ on $CE,BE,BC$ respectively, prove that lines $KA,DL$ and $ZN$ are concurrent.

2013 Cyprus TST 1.3

Given an equilateral triangle  $ABC$, On the sides $BC, AB$ we choose points $D,E$ respectively such that $BD=2AE$ and $\angle DEC=30^o$. If the perpendicular from $D$ on $EC$ intersects $EC$ at point $K$, prove that triangle $DKC$ is isosceles.

2013 Cyprus TST 2.3

Given an acute triangle  $ABC$. From point $D$ on its side $AB$ we draw parallel to $BC$, that intersects $AC$ at point $E$. Let $Z$ be a point of side $BC$. On the line $AZ$ we choose points $K, L$ (on the same halfplane wrt line $DE$) such that $\angle BKC=\angle DEC$ and $\angle DLE=\angle BCA$. If the circumscribed circles $(c_1), (c_2)$ of the triangles $DEL, BKC$ respectively intersect at points $M$ and $N$ , prove that line $MN$ passes through intersection point of lines $EL$ and $KC$.

2013 Cyprus TST 3.3

Given an acute triangle and non isosceles  $ABC$ with orthocenter $H$. The external bisector of angle $\angle BHC$ intersects the sides $AB, AC$ at points $D,E$ respectively. The bisector of angle $\angle BAC$ intersects the circumscribed circle $(C_1)$ of the triangle $ADE$ at point $K$ (different from $A$). If the line $HK$ intersects the circumscribed circle $(C_2)$ of the triangle $ABC$ at points $L$ and$N$, prove that triangle $ALN$ is right, with the vertex of the right angle to be the second intersection point of circles $(C_1), (C_2)$.

2013 Cyprus TST 4.3

Given a parallelogram $ABCD$ and a point $M$ on the diagonal $BD$. Line $AM$ intersects the lines $CD, BC$ at points $K,N$ respectively. Let $(c_1)$ be the circle with center $M$ and radius $MA$ .Let $(c_2)$ be the circumscribed circle of the triangle $KCN$. If $H, Z$ are the intersection points of the circles   $(c_1), (c_2)$ , prove that segments $MH$ and $MZ$ are tangent at circle $(c_2)$.

2014 Cyprus TST 1.3

In acute triangle $\vartriangle ABC$, we draw the altitude $AD$ and let $D_1,D_2$ be the symmetric points of $D$ wrt the sides $AB,AC$ respectively. If $D_1D_2$ interseects sides $AB,AC$ at points $E,Z$ respectively, prove that the segments $CE,BZ$ are altitudes of the triangle $\vartriangle ABC$.

2014 Cyprus TST 2.3

Given a triangle $\vartriangle ABC$ and let $(c_1)$ be it's circumscribed circle with center $O$. The line passing through vertex $A$ parallel to $BC$ intersects the circle as point $D$. Let $P$ be a point on the extension of $BC$ beyond $C$. Draw the circle with diameter the segment $PD$ and let $E$ be it's second intersection with circle $(c_1)$. Let $N= PA\cap (c_1)$, $T=PB\cap DE$, $H$ be the orthocenter of $\vartriangle ABC$ and $M$ be the midpoint of $BC$. Prove that lines $HM, AO$ and $NT$ are concurrent.

2014 Cyprus TST 3.3

Given a quadrilateral $ABCD$ inscirbed in circle $(O,r)$ and let $L,M$ be the  incenters of triangles $\vartriangle BCA$, $\vartriangle BCD$ respectively. Draw the perpendicular line $(e_1)$ from point $L$ on the line $AC$ and the perpendicular line $(e_2)$ from point $M$ on the line $BD$. Let $P$ be the intersection point of $(e_1)$ and $(e_2)$. Prove that the perpendicular from point $P$ on the line $LM$ passes though midpoint of segment $LM$.

2015 Cyprus TST 1.4

Given two circles $\omega_1, \omega_2$,  with $\omega_2$ to be bigger than $\omega_1$, internally tangent at point $A$. Let $B$ a point on $\omega_2$ and let $(e_1),(e_2)$ be the tangents from point $B$ to circle $\omega_1$. Let the line $(e_1)$ be tangent to $\omega_1$ at point $C$ and intersect circle $\omega_2$ again at point $D$. Also let the line $(e_2)$ be tangent to $\omega_1$ at point $Z$ and intersect circle $\omega_2$ again at point $E$. Let $M,N$ be the midpoints of arcs $BD, BE$  that do not contain $A$. If the circumscribed circles of the triangles $\vartriangle MBC$ and $\vartriangle BNZ$ intersect at point $H$, and $T,P$ are the midpoints of $MH,BN$ respectively, prove that lines $MN,BH,TP$ are concurrent.

2015 Cyprus TST 2.2

Given are two circles $K_1(O_1, r_1), K_2(O_1, r_1)$ such that $O_1O_2>r_1+r_2$. Let $(e_1),(e_2)$ be the common external tangents of the circles. Let $A,B$ be the intersection points of $(e_1)$ with circles $K_1, K_2$ respectively and let $C,D$ be the intersection points of $(e_2)$ with circles $K_1, K_2$ respectively. Also let $M$ be the midpoint of $CD$ and let $E,Z$ be the intersection points of segments $AM,BM$ with circles $K_1, K_2$ respectively.  Let $T, P$ be the intersection points of the line $EZ$ with circles $K_1, K_2$ respectively. Let the lines $AT$ and $BP$ intersect at point $K$. Let the tangnents of the circles at points $E,Z$ intersect at point $N$. Prove that the perpendiculars from points $O_1, O_2$ on segments $O_2M, O_1M$ respectively and line $KN$ are concurrent.

2015 Cyprus TST 3.3

Given a trapezoid $ABCD$ ($AB \parallel CD$) such that point $E$ is the intersection point of its diagonals and $EB=EC$. The perpendiculars from points $E$ and $B$ on the lines $BE$ and $BD$ respectively intersect at point $Z$. We suppose that the circumscribed circle of the triangle $\vartriangle BZD$ intersects line $AD$ at point $T$. If $H$ is the intersection point of $EZ$ with $BC$, prove that point $H,T,D,C$ are concyclic.

2016 Cyprus TST 1.2

Given isosceles triangle $\vartriangle ABC$ ($AB=AC$), $AE$ it's altitude and let $(k)$ be the circumscribed circle of the triangle, which has center $O$. Draw the tangent $(e)$ of circle $(k)$ at point $B$. From point $A$ draw parallel to $(e)$, that intersects line $BC$ at point $D$. If $O'$ is the symmetric point of $O$ wrt$BC$ and $T$ is the intersection point of $AE$ with circle $(k)$, prove that:

a) $AC$ is perpendicular to $O'D$

b) $OC \cdot EC = ET \cdot AD$

2016 Cyprus TST 2.3

From vertex $A$ of a scalene tirangle $ABC$, we draw the line $(e) \parallel BC$, line $(u) \perp BC$ and the bisector $(d)$ of angle $\angle BAC$. From the midpoint $M$ of side $BC$, we draw perpendicular on $(d)$, that intersects $(d)$ at point $H$, $(u)$ at point $E$ and $(e)$ at point $Z$. Prove that $BM^2=MH \cdot ZE$.

2016 Cyprus TST 3.4

Quadrilateral $ABCD$ is inscribed in circle $(\omega)$ with center $O$, $\angle B =\angle D=90^o$ and $AB=AD<BC$. Let $X$ be a point on $BD$. Line $AX$ intersects again circle $(\omega)$ at point $S$ different from A. From the point $X$ we draw perpendicular on $AS$ that intersects arc $ADC$ at point $T$. Let $M,Z$ be the midpoints of $ST, AO$ respectively. Prove that $BZ=ZM$.

2017 Cyprus TST 1.4

Given an acute triangle $\vartriangle ABC$ with altitudes $AD, BE, CZ$ and $H$ the intesection point of it's altitudes. Draw line $(e)$ passing through points $Z$ and $D$. On line $(e)$, mark points $K$ and $L$ such that altitude $CZ$ pass through midpoint of segment $KE$ and altitude $AD$ pass through midpoint of segment $LE$. If $M$ is the intersection point of the lines $KE, AD$ and $N$ is the intersection point of lines $LE, CZ$, prove that:

i) $\angle NMA=\angle EBC$

ii) points $N, L, H, K, M$ are concyclic

iii) the second intersection point of the circumscribed circles of the triangles  $\vartriangle NLH, \vartriangle ECH$ is point $K$.

2017 Cyprus TST 2.3

Let $ABC$ be an acute triangle $\vartriangle ABC$ with $AB< AC$ and $H$ it's orthocenter. From $H$ draw line perpendicular to bisector $(d)$ of angle $\angle BAC$ of the triangle, that intersects it's sides $AB, AC$ at point $T, I$ respectively. Denote $X$ the second intersection point of the circumscribed circles $(c_1),(c_2)$ ot the triangles $\vartriangle ATI$, $\vartriangle ABC$ respectively. Let $L$ be the intersection point of $(c_1)$ with the angle bisector $(d)$, different than $A$, and let N be the intersection point of the line $HL$ with $BC$. If $D$ is the foot of the altitude from vertex $A$ on the side $BC$, prove that $\angle BDX=\angle XAN$.

2017 Cyprus TST 3.2

Let $ABC$ be an acute triangle $\vartriangle ABC$ with $AB< AC$ and let $D, E$ be the feet of the perpendicular from points $B,C$ on the opposite sides respectively. Let $S, T$ be the symmetrics of point $E$ wrt sides $AC, BC$ respectively. Suppose that the circumscribed circle of the triangle  $CST$ has center $O$ and intersects line $AC$ at point $Z \ne C$. Let $N$ be the circumcenter of $\vartriangle ABC$. Prove that lines $ZO$ and $AN$ are parallel.

2018 Cyprus TST 1.4

Given an acute isosceles triangle $\vartriangle ABC$ ($AB=AC$) and let $O$ be it's circumcenter. Lines $BO$ and $AO$ intersect the sides $CA,CB$ at points $D,E$ respectively. From the point $D$, we draw line $(e)$ parallel to $CB$ that intersects $AB$ at point $Z$. Let $H$ be the intersection point of the perpendicular bisector of $BE$ with line $(e)$. Draw the circle $(C,CH)$ and denote $(t_1)$ the semicircle of those circle, that lies on the same halfplane defines by $CH$ with vertex $B$. If the tangents from the points $A,B$ are tangent to $(t_1)$ at points $K,L$ respectively, and $M$ is the midpoint of $AB$, prove that $\angle MHZ=\angle AKM=\angle KLB$.

2018 Cyprus TST 2.3

Given two circles $c_1 (O,R_1$)  and $c_2 (K,R_2)$ with $R_2>R_1$, that are externally tangent at point $M$. From a point $A$ of the circle $c_2$ that does not les on the line $OK$, we draw the tangents $(e_1), (e_2)$ to the circle $c_1$ and let $B,C$ be the touch point respectively with the circle $c_1$. Lines $MB, MC$ intersect circle $c_2$ again at the points $E,Z$ respectively. Let $L$ be the intersection point of line $EZ$ and the tangent of the circle $c_2$ at the point $A$. Prove that $LM \perp OK$.

2018 Cyprus TST 3.2

Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.

2019 Cyprus TST 1.3

Let $ABCD$ be a quadrilateral inscribed in circle with center $O$. Denote by $T$ the intersection point of the bisectors $(d_1),(d_2)$ of the angles $\angle CAD, \angle CBD$ respectively. Let $A',B'$ be the symmetric points of $A,B$ wrt $(d_2),(d_1)$ respectively. Let $P$ be the intersection point of $AB'$ and $BA'$. Prove that points $T, O, P$ are collinear. 

2019 Cyprus TST 2.3

Given an acute triangle $\vartriangle ABC$ inscribed in circle with center $O$, and $AD$ it's altitude. Let $T$ be the intersection point of $CO$ with $AD$. Let $N,M$ be the midpoints of segments $AT, AC$ respectively. Let $X$ be the intersection point of line $NO$ with $BC$. If $K,O_1$ are the centers of the circumscribed of the triangles $\vartriangle BCM, \vartriangle BOX$ respectively, prove thar $KO_1 \perp AB$.

2019 Cyprus TST 3.2

Two circles $\omega_1, \omega_2$ intersect at points $A,B$. Random line passing through $B$, intersects  $\omega_1, \omega_2$ at points $C,D$ respectively. Choose points $E,Z$ on  $\omega_1, \omega_2$ respectively such that $CE=CB, BD=DZ$. Suppose the $BZ$ intersects $\omega_1$ at $P$. Prove that line $AP$ is bisector of angle $\angle EAZ$.

2020 Cyprus TST 1.3

Given a triangle $ABC$ inscribed in circle $\omega$. Perpendicular on $AC$ at point $A$, intersects again $\omega$ at point $M$ . Perpendicular on $AB$ at point $A$, intersects again $\omega$ at point $N$. The perpendiculars from points $M,N$ on lines $AB,AC$ respectively intersect at point $H$. Let $T$ be the intersection point of lines $AH$ and $MN$. If the bisector of angle $\angle BAC$ intersects the circle at point $P$, prove that line $TP$ passes through the misdpoint of segment $BC$.

2020 Cyprus TST 2.3

Given a triangle $ABC$ with $AC>BC>AB$ and orthocenter $H$.  Let $(c)$ be the circumscribed circle of vartriangle $ABC$ and $O$ the circumcenter. The tangent of $(c)$ at A, intersects $BC$ at $D$. Let $Z$ be the midpoint of $AH$. The perpendicular from $O$ on $DZ$ intersects $DZ$ at point $T$. Let $M,N$ be the midpoints of sides $BC, AB$ respectively. If $K$ is the midpoint of $ZT$, prove that $\angle ZNK= \angle ZMK$.

2021 Cyprus TST 1.4

Given a triangle $ABC$ with $AB \ne AC$ and $D,E,Z$ the midpoints of the sides $BC$, $AC$ and $AB$ respectively. With diameters the segments $AB$ and $AC$, we draw semicircles outside the triangle. Lines $DE$ and $DZ$ intersect the semicircles at point $T$ and $H$ respectively. Tangents of the semicircles at points $T$ and $H$ intersect at $I$. $DA$ intersects the circumscribed circle of the triangle $HDT$ at point $X$. Prove that lines $IX$ and $HT$ intersect on line $BC$.

2021 Cyprus TST 2.3

Given an acute triangle $ABC$ wit $AB < AC$ and let $M$ be the midpoint of $BC$. From point $M$, we draw a line intersecting the lines $AB$ and $AC$ at points $I$ and $K$ respectively, such that $AI=AK$. Let $O$ be the center of the circumscribed circle of triangle $AIK$, and $D$ be the foot of the perpendicular from point $A$ on $BC$. Prove that triangle $ODM$ is isosceles.

source: https://www.cms.org.cy/pages/competitions/competition-cms/selection-competitions-for-imo-bmo-egmo-mymc