Moldova TST (IMO - EGMO) 2002-19 38p (UC)

geometry problems from Moldovan Team Selection Tests (TST)
with aops links in the names
(only those not in IMO Shortlist)
[4p per day]

2002 - 2019
(missing 2004, 2012, 2016 so far)

2002 Moldova TST P3
The circles $G_1(O_1), G_2(O_2), G_3(O_3)$ are such that $G_1$ and $G_2$ are externally tangent at $A, G_2,G_3$ are so at $B$, and $G_3,G_1$ are so at $C$. Let $A_1$ and $B_1$ be points on $G_1$ diametrically opposite to $A$ and $B$ respectively, and let $AB_1$ meet $G_2$ again at $M, BA_1$ meet $G_3$ again at $N$, and $AA_1$ and $BB_1$ meet at $P$. Prove that points $M,N,P$ are collinear.

2002 Moldova TST P7
A triangle $ABC$ is inscribed in a circle $G$. Points $M$ and $N$ are the midpoints of the arcs $BC$ and $AC$ respectively, and $D$ is an arbitrary point on the arc $AB$ (not containing $C$). Points $I_1$ and $I_2$ are the incenters of the triangles $ADC$ and $BDC$, respectively. If the circumcircle of triangle $DI_1I_2$ meets $G$ again at $P$, prove that triangles $PNI_1$ and $PMI_2$ are similar.

A triangle $ABC$ is inscribed in a circle $G$. For any point $M$ inside the triangle, $A_1$ denotes the intersection of the ray $AM$ with $G$. Find the locus of point M for which $\frac{BM\cdot CM}{MA_1}$ is minimal, and find this minimum value.

2003 Moldova TST P3
Let $ ABCD$ be a quadrilateral inscribed in a circle of center $ O$. Let M and N be the midpoints of diagonals $ AC$ and $ BD$, respectively and let $ P$ be the intersection point of the diagonals $ AC$ and $ BD$ of the given quadrilateral .It is known that the points $ O,M,Np$ are distinct. Prove that the points $ O,N,A,C$ are concyclic if and only if the points $ O,M,B,D$ are concyclic.

by Dorian Croitoru
2003 Moldova TST P7
The sides $ [AB]$ and $ [AC]$ of the triangle $ ABC$ are tangent to the incircle with center $ I$ of the $ \triangle ABC$ at the points $ M$ and $ N$, respectively. The internal bisectors of the $ \triangle ABC$ drawn form $ B$ and $ C$ intersect the line $ MN$ at the points $ P$ and $ Q$, respectively. Suppose that $ F$ is the intersection point of the lines $ CP$ and $ BQ$. Prove that $ FI\perp BC$.

2003 Moldova TST P11
Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation: $ (MA^2+MB^2+MC^2)^2=16(S_1^2+ S_2^2 +S_3^2)$

2004 Moldova TST P
In acute $\triangle ABC$ $H$ and $O$ are the orthocentre and the circumcentre respectively.Prove that for any point $P$ on the segment $OH$ we have $6r \leq AP+BP+CP$ where $r$ is the inradius for $ABC$

missing 2004

2005 Moldova TST P5
In triangle $ABC$, $M\in(BC)$, $\frac{BM}{BC}=\alpha$, $N\in(CA)$, $\frac{CN}{CA}=\beta$, $P\in(AB)$, $\frac{AP}{AB}=\gamma$. Let $AM\cap BN=\{D\}$, $BN\cap CP=\{E\}$, $CP\cap AM=\{F\}$. Prove that $S_{DEF}=S_{BMD}+S_{CNE}+S_{APF}$ iff $\alpha+\beta+\gamma=1$.

2006 Moldova TST P2
Consider a right-angled triangle $ABC$ with the hypothenuse $AB=1$. The bisector of $\angle{ACB}$ cuts the medians $BE$ and $AF$ at $P$ and $M$, respectively. If ${AF}\cap{BE}=\{P\}$, determine the maximum value of the area of $\triangle{MNP}$.

2006 Moldova TST P6
Let $C_1$ be a circle inside the circle $C_2$ and let $P$ in the interior of $C_1$, $Q$ in the exterior of $C_2$. One draws variable lines $l_i$ through $P$, not passing through $Q$. Let $l_i$ intersect $C_1$ in $A_i,B_i$, and let the circumcircle of $QA_iB_i$ intersect $C_2$ in $M_i,N_i$. Show that all lines $M_i,N_i$ are concurrent.

2006 Moldova TST P9
Let the point $P$ in the interior of the triangle $ABC$. $(AP, (BP, (CP$ intersect the circumcircle of $ABC$ at $A_{1}, B_{1}, C_{1}$. Prove that the maximal value of the sum of the areas $A_{1}BC$, $B_{1}AC$, $C_{1}AB$ is $p(R-r)$, where $p, r, R$ are the usual notations for the triangle $ABC$.

2007 Moldova TST P1
Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.

2007 Moldova TST P3
Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point.

Remark. The Fermat point $F$ is also known as the first Fermat point or the first Toricelli point of triangle $ABC$.

Floor van Lamoen

2007 Moldova TST P7
Let $M, N$ be points inside the angle $\angle BAC$ such that $\angle MAB\equiv \angle NAC$. If $M_{1}, M_{2}$ and $N_{1}, N_{2}$ are the projections of $M$ and $N$ on $AB, AC$ respectively then prove that $M, N$ and $P$ the intersection of $M_{1}N_{2}$ with $N_{1}M_{2}$ are collinear.

2007 Moldova TST P11
Consider a triangle $ABC$, with corresponding sides $a,b,c$, inradius $r$ and circumradius $R$. If $r_{A}, r_{B}, r_{C}$ are the radii of the respective excircles of the triangle, show that
\[a^{2}\left(\frac 2{r_{A}}-\frac{r}{r_{B}r_{C}}\right)+b^{2}\left(\frac 2{r_{B}}-\frac{r}{r_{A}r_{C}}\right)+c^{2}\left(\frac 2{r_{C}}-\frac{r}{r_{A}r_{B}}\right)=4(R+3r) \]

2007 Moldova TST P15
Let $ABC$ be a triangle. A circle is tangent to sides $AB, AC$ and to the circumcircle of $ABC$ (internally) at points $P, Q, R$ respectively. Let $S$ be the point where $AR$ meets $PQ$. Show that $\angle{SBA}\equiv \angle{SCA}$

2008 Moldova TST P3
Let $ \Gamma(I,r)$ and $ \Gamma(O,R)$ denote the incircle and circumcircle, respectively, of a triangle $ ABC$. Consider all the triangels $ A_iB_iC_i$ which are simultaneously inscribed in $ \Gamma(O,R)$ and circumscribed to $ \Gamma(I,r)$. Prove that the centroids of these triangles are concyclic.

2008 Moldova TST P7
Let $ \omega$ be the circumcircle of $ ABC$ and let $ D$ be a fixed point on $ BC$, $ D\neq B$, $ D\neq C$. Let $ X$ be a variable point on $ (BC)$, $ X\neq D$. Let $ Y$ be the second intersection point of $ AX$ and $ \omega$. Prove that the circumcircle of $ XYD$ passes through a fixed point.

2008 Moldova TST P11
In triangle $ ABC$ the bisector of $ \angle ACB$ intersects $ AB$ at $ D$. Consider an arbitrary circle $ O$ passing through $ C$ and $ D$, so that it is not tangent to $ BC$ or $ CA$. Let $ O\cap BC = \{M\}$ and $ O\cap CA = \{N\}$.
a) Prove that there is a circle $ S$ so that $ DM$ and $ DN$ are tangent to $ S$ in $ M$ and $ N$, respectively.
b) Circle $ S$ intersects lines $ BC$ and $ CA$ in $ P$ and $ Q$ respectively. Prove that the lengths of $ MP$ and $ NQ$ do not depend on the choice of circle $ O$.

2009 Moldova TST P3
Quadrilateral $ ABCD$ is inscribed in the circle of diameter $ BD$. Point $ A_1$ is reflection of point $ A$ wrt $ BD$ and $ B_1$ is reflection of $ B$ wrt $ AC$. Denote $ \{P\}=CA_1 \cap BD$ and $ \{Q\}=DB_1\cap AC$. Prove that $ AC\perp PQ$.

A circle $ \Omega_1$ is tangent outwardly to the circle $ \Omega_2$ of bigger radius. Line $ t_1$ is tangent at points $ A$ and $ D$ to the circles $ \Omega_1$ and $ \Omega_2$ respectively. Line $ t_2$, parallel to $ t_1$, is tangent to the circle $ \Omega_1$ and cuts $ \Omega_2$ at points $ E$ and $ F$. Point $ C$ belongs to the circle $ \Omega_2$ such that $ D$ and $ C$ are separated by the line $ EF$. Denote $ B$ the intersection of $ EF$ and $ CD$. Prove that circumcircle of $ ABC$ is tangent to the line $ AD$.

2009 Moldova TST P9
Points $ X$, $ Y$ and $ Z$ are situated on the sides $ (BC)$, $ (CA)$ and $ (AB)$ of the triangles $ ABC$, such that triangles $ XYZ$ and $ ABC$ are similiar. Prove that circumcircle of $ AYZ$ passes through a fixed point.

2009 Moldova TST P13
Let $ ABCD$ be a trapezoid with $ AB\parallel CD$. Exterior equilateral triangles $ ABE$ and $ CDF$ are constructed. Prove that lines $ AC$, $ BD$ and $ EF$ are concurrent.

2010 Moldova TST P3
Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC=3\angle CAD$, $ AB=AD$, $ \angle ACD=\angle CBD$. Find angle $ \angle ACD$

2010 Moldova TST P7
Let $ ABC$ be an acute triangle. $ H$ is the orthocenter and $ M$ is the middle of the side $ BC$. A line passing through $ H$ and perpendicular to $ HM$ intersect the segment $ AB$ and $ AC$ in $ P$ and $ Q$. Prove that $ MP = MQ$

2011 Moldova TST P3
Let $ABCD$ be a quadrilateral and $M$ the midpoint of the segment $AB$. Outside of the quadrilateral are constructed the equilateral triangles $BCE$, $CDF$ and $DAG$. Let $P$ and $N$ be the midpoints of the segments $GF$ and $EF$. Prove that the triangle $MNP$ is equilateral.

2011 Moldova TST P7
Let $ABC$ be a triangle with $\angle BAC=60$. Let $B_1$ and $C_1$ be the feet of the bisectors from $B$ and $C$. Let $A_1$ be the symmetrical of $A$ according to line $B_1C_1$. Prove that $A_1, B, C$ are colinear.

2012 Moldova TST P
Let $C(O_1),C(O_2)$ be two externally tangent circles at point $P$. A line $t$ is tangent to $C(O_1)$ in point $R$ and intersects $C(O_2)$ in points $A,B$ such that $A$ is closer to $R$ than $B$ is. The line $AO_1$ intersects the perpendicular to $t$ in $B$ at point $C$, the line $PC$ intersects $AB$ in $Q$.  Prove that $QO_1$ passes through the midpoint of $BC$.

missing 2012 [no geometry not from ISL in 2013]

Let $\triangle ABC$ be an acute triangle and $AD$ the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let $E$ and $F$ denote feet of perpendiculars from $D$ to $AB$ and $AC$ respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.

2014 Moldova TST P7
Let $ABCD$ be a cyclic quadrilateral. The bisectors of angles $BAD$ and $BCD$ intersect in point $K$ such that $K \in BD$. Let $M$ be the midpoint of $BD$. A line passing through point $C$ and parallel to $AD$ intersects $AM$ in point $P$. Prove that triangle $\triangle DPC$ is isosceles.

Let $\triangle ABC$ be a triangle with $\angle A$-acute. Let $P$ be a point inside $\triangle ABC$ such that $\angle BAP = \angle ACP$ and $\angle CAP =\angle ABP$. Let $M, N$ be the centers of the incircle of $\triangle ABP$ and $\triangle ACP$, and $R$ the radius of the circumscribed circle of $\triangle AMN$. Prove that $\displaystyle \frac{1}{R}=\frac{1}{AB}+\frac{1}{AC}+\frac{1}{AP}. $

2015 Moldova TST P2
Consider a triangle $\triangle ABC$, let the incircle centered at $I$ touch the sides $BC,CA,AB$ at points $D,E,F$ respectively. Let the angle bisector of $\angle BIC$ meet $BC$ at $M$, and the angle bisector of $\angle EDF$ meet $EF$ at $N$. Prove that $A,M,N$ are collinear.

2015 Moldova TST P7
Consider an acute triangle $ABC$, points $E,F$ are the feet of the perpendiculars from $B$ and $C$ in $\triangle ABC$. Points $I$ and $J$ are the projections of points $F,E$ on the line $BC$, points $K,L$ are on sides $AB,AC$ respectively such that $IK \parallel AC$ and $JL \parallel AB$. Prove that the lines $IE$,$JF$,$KL$ are concurrent.

missing 2016

2017 Moldova TST P3
Let $\omega$ be the circumcircle of the acute nonisosceles triangle $\Delta ABC$. Point $P$ lies on the altitude from $A$. Let $E$ and $F$ be the feet of the altitudes from P to $CA$, $BA$ respectively. Circumcircle of triangle $\Delta AEF$ intersects the circle $\omega$ in $G$, different from $A$. Prove that the lines $GP$, $BE$ and $CF$ are concurrent.

2018 Moldova TST P7
Let the triangle $ABC $ with $m (\angle ABC)=60^{\circ} $ and $m (\angle BAC)=40^{\circ}$ . Points $D $ and $E $ are on the sides $(AB) $ and $(AC) $ such that $m (\angle DCB )=70^{\circ}$ and $m (\angle EBC)=40^{\circ}$ . $BE$ and $CD$ intersect in  $F $ . Prove that $BC $ and $AF $ are perpendicular.

2018 Moldova TST P11
Let $\Omega $ be the circumcincle of the quadrilater $ABCD $ , and $E $ the intersection point of the diagonals  $AC $ and $BD $ . A line passing through  $E $ intersects  $AB $ and $BC$ in points $P $ and $Q $ . A circle ,that is passing through point $D $ , is tangent to the line $PQ $ in point $E $ and intersects $\Omega$ in point $R $ , different from $D $ . Prove that the points  $B,P,Q,$ and $R $ are concyclic .

2019 Moldova TST P4
Quadrilateral $ABCD$ is inscribed in circle $\Gamma$ with center $O$. Point $I$ is the incenter of triangle $ABC$, and point $J$ is the incenter of the triangle $ABD$. Line $IJ$ intersects segments $AD, AC, BD, BC$ at points $P, M, N$ and, respectively $Q$. The perpendicular from $M$ to line $AC$ intersects the perpendicular from $N$ to line $BD$ at point $X$. The perpendicular from $P$ to line $AD$ intersects the perpendicular from $Q$ to line $BC$ at point $Y$. Prove that $X, O, Y$ are colinear.

2019 Moldova TST P5
Point $H$ is the orthocenter of the scalene triangle $ABC.$ A line, which passes through point $H$, intersect the sides $AB$ and $AC$ at points $D$ and $E$, respectively, such that $AD=AE.$ Let $M$ be the midpoint of side $BC.$ Line $MH$ intersects the circumscribed circle of triangle $ABC$ at point $K$, which is on the smaller arc $AB$. Prove that Nibab can draw a circle through $A, D, E$ and $K.$

2019 Moldova TST P10
Circle $\Omega$ with center $O$ is the circumcircle of the acute angled triangle $ABC.$ Let $P$ be an arbitrary point of the circumcircle of triangle $OBC$ such that $P$ is in the interior of triangle $ABC$ and is different from $B$ and $C.$ The bisectors of angles $NPA$ and $CPA$ intresect the sides $AB$ and $AC$ at point $E$ and $F$, respectively. Prove that the incenters of triangle $PEF, PCA$ and $PBA$ are colinear.

EGMO TST 2017-18 

Let us denote the midpoint of $AB$ with $O$. The point $C$, different from $A$ and $B$ is on the circle $\Omega$ with center $O$ and radius $OA$ and the point $D$ is the foot of the perpendicular from $C$ to $AB$. The circle with center $C$ and radius $CD$ and $\omega$ intersect at $M$, $N$. Prove that $MN$ cuts $CD$ in two equal segments.

The points $P$ and $Q$ are placed in the interior of the triangle $\Delta ABC$ such that $m(\angle PAB)=m(\angle QAC)<\frac{1}{2}m(\angle BAC)$ and similarly for the other $2$ vertices($P$ and $Q$ are isogonal conjugates). Let $P_{A}$ and $Q_{A}$ be the intersection points of $AP$ and $AQ$ with the circumcircle of $CPB$, respectively $CQB$. Similarly the pairs of points $(P_{B},Q_{B})$ and $(P_{C},Q_{C})$ are defined. Let $PQ_{A}\cap QP_{A}=\{M_{A}\}$, $PQ_{B}\cap QP_{B}=\{M_{B}\}$, $PQ_{C}\cap QP_{C}=\{M_{C}\}$.
Prove the following statements:
$1.$ Lines $AM_{A}$, $BM_{B}$, $CM_{C}$ concur.
$2. $ $M_{A}\in BC$, $M_{B}\in CA$, $M_{C}\in AB$

Let $\triangle ABC $ be an acute triangle.$O$ denote its circumcenter.Points $D$,$E$,$F$ are the midpoints of the sides $BC$,$CA$,and $AB$.Let $M$ be a point on the side $BC$  .  $ AM \cap EF = \big\{ N \big\} $ .  $ON \cap \big( ODM \big) = \big\{ P  \big\} $ Prove that $M'$ lie on $\big(DEF\big)$ where $M'$ is the symmetrical point of $M$ thought the midpoint of $DP$.

Let $ABCD$ be a isosceles trapezoid with $AB  \| CD $ , $AD=BC$, $ AC \cap BD = $ { $O$ }. $ M $ is the midpoint of the side $AD$ . The circumcircle of triangle $ BCM $ intersects again the side $AD$ in $K$. Prove that $OK  \|  AB $ . 

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