Moldova TST 2002-20 (IMO - EGMO) 44p (-04,-12)

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

2002 - 2020
(missing 2004, 2012 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.

2016 Moldova TST P7 (ELMO SL 2015, China TST 2016)
Let $\Omega$ and $O$ be the circumcircle of acute triangle $ABC$ and its center, respectively. $M\ne O$ is an arbitrary point in the interior of $ABC$ such that $AM$, $BM$, and $CM$ intersect $\Omega$ at $A_{1}$, $B_{1}$, and $C_{1}$, respectiuvely. Let $A_{2}$, $B_{2}$, and $C_{2}$ be the circumcenters of $MBC$, $MCA$, and $MAB$, respectively. It is to be proven that $A_{1}A_{2}$, $B_{1}B_{2}$, $C_{1}C{2}$ concur.

Let $A_{1}A_{2} \cdots A_{14}$ be a regular $14-$gon. Prove that $A_{1}A_{3}\cap A_{5}A_{11}\cap A_{6}A_{9}\ne \emptyset$.

Let $ABCD$ be a cyclic quadrilateral. Circle with diameter $AB$ intersects $CA$, $CB$, $DA$, and $DB$ in $E$, $F$, $G$, and $H$, respectively (all different from $A$ and $B$). The lines $EF$ and $GH$ intersect in $I$. Prove that the bisector of $\angle GIF$ and the line $CD$ are perpendicular.

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.

2020 Moldova TST P4
Let $\Delta ABC$ be an acute triangle and $H$ its orthocenter. $B_1$ and $C_1$ are the feet of heights from $B$ and $C$, $M$ is the midpoint of $AH$. Point $K$ is on the segment $B_1C_1$, but isn't on line $AH$. Line $AK$ intersects the lines $MB_1$ and $MC_1$ in $E$ and $F$, the lines $BE$ and $CF$ intersect at $N$. Prove that $K$ is the orthocenter of $\Delta NBC$.

2020 Moldova TST P8
In $\Delta ABC$ the angles $ABC$ and $ACB$ are acute. Let $M$ be the midpoint of $AB$. Point $D$ is on the half-line $(CB$ such that $B \in (CD)$ and $\angle DAB= \angle BCM$. Perpendicular from $B$ to line $CD$ intersects the line bisector of $AB$ in $E$. Prove that $DE$ and $AC$ are perpendicular.

Let $\Delta ABC$ be an acute triangle and $\Omega$ its circumscribed circle, with diameter $AP$. Points $E$ and $F$ are the orthogonal projections from $B$ on $AC$ and $AP$, points $M$ and $N$ are the midpoints of segments $EF$ and $CP$. Prove that $\angle BMN=90$.

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 $OK \| AB$ .