Moldova JBMO TST 1999 - 2022 (-09) 45p

geometry problems from Moldovan Junior Balkan Mathematical Olympiads Team Selection Tests (JBMO TST) with aops links

[not in JBMO Shortlist]

collected inside aops here

1999 - 2022
2009 missing

1999 Moldova  JBMO TST 1.2

Let $ABC$ be an isosceles right triangle with $\angle A=90^o$. Point $D$ is the midpoint of the side $[AC]$, and point $E \in [AC]$ is so that $EC = 2AE$. Calculate $\angle AEB + \angle ADB$ .

1999 Moldova JBMO TST 1.4

Let $ABC$ be an equilateral triangle of area $1998$ cm$^2$. Points $K, L, M$ divide the segments $[AB], [BC] ,[CA]$, respectively, in the ratio $3:4$ . Line $AL$ intersects the lines $CK$ and $BM$ respectively at the points $P$ and $Q$, and the line $BM$ intersects the line $CK$ at point $R$. Find the area of the triangle $PQR$.

2000 Moldova JBMO TST 1.3

Let $ABC$ be a triangle with $AB = AC$ ¸ $\angle BAC  = 100^o$ and $AD, BE$ angle bisectors. Prove that $2AD <BE + EA$

2000 Moldova  JBMO TST 2.3

Let a triangle $ABC, A_1$ be the midpoint of the segment $[BC], B_1 \in (AC)$ ¸and $C_1 \in  (AB)$ such that $[A_1B_1$ is the bisector of the angle $AA_1C$ and $A_1C_1$ is perpendicular to $AB$. Show that the lines $AA_1, BB_1$ and $CC_1$ are concurrent if and only if $\angle  BAC = 90^o$ 

2001 Moldova  JBMO TST 1.3

Let the convex quadrilateral $ABCD$ with $AD = BC$ ¸and $\angle A + \angle B = 120^o$. Take a point $P$ in the plane so that the line $CD$ separates the points $A$ and $P$, and the $DCP$ triangle is equilateral. Show that the triangle $ABP$ is equilateral. It is the true statement for a non-convex quadrilateral?

2002 Moldova JBMO TST 1.3

Let $ABC$ be a an acute triangle. Points $A_1, B_1$ and $C_1$ are respectively the projections of the vertices $A, B$ and $C$ on the opposite sides of the triangle, the point $H$ is the orthocenter of the triangle, and the point $P$ is the middle of the segment $[AH]$. The lines $BH$ and $A_1C_1$, $P B_1$ and $AB$ intersect respectively at the points $M$ and $N$. Prove that the lines $MN$ and $BC$ are perpendicular.

2002 Moldova  JBMO TST 2.3
The side of the square $ABCD$ has a length equal to $1$. On the sides $(BC)$ ¸and $(CD)$ take respectively the arbitrary points $M$ and $N$ so that the perimeter of the triangle $MCN$ is equal to $2$.
a) Determine the measure of the angle $\angle MAN$.
b) If the point $P$ is the foot of the perpendicular taken from point $A$ to the line $MN$, determine the locus of the points $P$.

2002 Moldova JBMO TST 3.2
The circles $C_1$ and $C_2$ intersect at the distinct points $M$ and $N$. Points $A$ and $B$ belong respectively to the circles $C_1$ and $C_2$ so that the chords $[MA]$ and  $[MB]$ are tangent at point $M$ to the circles $C_2$ and $C_1$, respectively. To prove it that the angles $\angle MNA$ and $\angle MNB$ are equal.

2003 Moldova JBMO TST 1.3
The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$

2003 Moldova JBMO TST 2.3
The triangle $ABC$ is isosceles with $AB=BC$. The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$. Fine the measure of the angle $ABC$.

2004 Moldova JBMO TST 1.3
Let $ABCD$ be a parallelogram  and point $M$ be the midpoint of $[AB]$ so that the quadrilateral $MBCD$ is cyclic. If $N$ is the point of intersection of the lines $DM$ and $BC$,  and $P \in  BC$, then prove that the ray $(DP$ is the angle bisector of $\angle ADM$ if and only if $PC = 4BC$.

2004 Moldova JBMO TST 2.3
Let the triangle $ABC$ have area $1$. The interior bisectors of the angles $\angle BAC,\angle ABC, \angle BCA$ intersect the sides $(BC), (AC), (AB)$ and the circumscribed circle of the respective triangle $ABC$ at the points $L$ and $G, N$ and $F, Q$ and $E$. The lines $EF, FG,GE$ intersect the bisectors $(AL), (CQ) ,(BN)$ respectively at points $P, M, R$. Determine the area of the hexagon $LMNPR$.

2005 Moldova JBMO TST 1.1
Let the triangle $ABC$ with $BC$ the smallest side. Let $P$ on ($AB$) such that angle $PCB$ equals angle $BAC$. and $Q$ on side ($AC$) such that angle $QBC$ equals angle $BAC$. Show that the line passing through the circumenters of triangles $ABC$ and $APQ$ is perpendicular on $BC$.

2005 Moldova JBMO TST 2.1
Let $ABC$ be an acute-angled triangle, and let $F$ be the foot of its altitude from the vertex $C$. Let $M$ be the midpoint of the segment $CA$. Assume that $CF=BM$. Then the angle $MBC$ is equal to angle $FCA$ if and only if the triangle $ABC$ is equilateral.

2006 Moldova JBMO TST 1.1
Five segments have lengths such that any three of them can be sides of a - possibly degenerate - triangle. Also, the lengths of these segments are nonzero and pairwisely different. Prove that there exists at least one acute-angled triangle among these triangles.

2006 Moldova JBMO TST 1.3
The convex polygon $A_{1}A_{2}\ldots A_{2006}$ has opposite sides parallel $(A_{1}A_{2}||A_{1004}A_{1005}, \ldots)$.
Prove that the diagonals $A_{1}A_{1004}, A_{2}A_{1005}, \ldots A_{1003}A_{2006}$ are concurrent if and only if opposite sides are equal.

2006 Moldova JBMO TST 2.2
Let $ABCD$ be a rectangle and denote by $M$ and $N$ the midpoints of $AD$ and $BC$ respectively. The point $P$ is on $(CD$ such that $D\in (CP)$, and $PM$ intersects $AC$ in $Q$. Prove that $m(\angle{MNQ})=m(\angle{MNP})$.

2007 Moldova JBMO TST 1.3
Let $ABC$ be a triangle with $BC = a, AC = b$ and $AB = c$. A point $P$ inside the triangle has the property that for any line passing through $P$ and intersects the lines $AB$ and $AC$ in the distinct points $E$ and $F$ we have the relation $\frac{1}{AE} +\frac{1}{AF} =\frac{a + b + c}{bc}$. Prove that the point $P$ is the center of the circle inscribed in the triangle $ABC$.

2008 Moldova JBMO TST 1.3
Rhombuses $ABCD$ and $A_1B_1C_1D_1$ are equal. Side BC intersects sides $B_1C_1$ and $C_1D_1$ at points $M$ and $N$ respectively. Side $AD$ intersects sides $A_1B_1$ and $A_1D_1$ at points $Q$ and $P$ respectively. Let $O$ be the intersection point of lines $MP$ and $QN$. Find $\angle A_1B_1C_1$ , if  $\angle  QOP  = \frac12 \angle B_1C_1D_1$.

2008 Moldova JBMO TST 2.3
In an acute triangle $ABC$, points $A_1, B_1, C_1$ are the midpoints of the sides $BC, AC, AB$, respectively. It is known that $AA_1 = d(A_1, AB) + d(A_1, AC)$, $BB1 = d(B_1, AB) + d(A_1, BC)$, $CC_1 = d(C_1, AC) + d(C_1, BC)$, where $d(X, Y Z)$ denotes the distance from point $X$ to the line $YZ$. Prove, that triangle $ABC$ is equilateral.

2008 Moldova JBMO TST 3.3 [typo exists]
Let $ABCD$ be a convex quadrilateral with $AD = BC, CD \nparallel  AB, AD \nparallel BC$. Points $M$ and $N$ are the midpoints of the sides $CD$ and $AB$, respectively.
a) If $E$ and $F$ are points, such that $MCBF$ and $ADME$ are parallelograms, prove that $\vartriangle  BF N \equiv \vartriangle  AEN$.
b) Let $P = MN  \cap  BC$, $Q = AD  \cap  MN$, $R = AD \cap BC$. Prove that the triangle $PQR$ is iscosceles.

2009 missing

2010 Moldova JBMO TST 1.3

The tangent to the circle circumscribed to the triangle $ABC$, taken through the vertex $A$, intersects the line $BC$ at the point $P$, and the tangents to the same circle, taken through $B$ and $C$, intersect the lines $AC$ and $AB$, respectively at the points $Q$ and $R$. Prove that the points $P, Q$ ¸ and $R$ are collinear.

2011 Moldova JBMO TST 1.3

Let $ABC$ be a triangle with $\angle ACB  = 90^o + \frac12 \angle ABC$ . The point $M$ is the midpoint of the side $BC$ . A circle with center at vertex $A$ intersects the line $BC$ at points $M$ and $D$. Prove that $MD = AB$.

2011 Moldova JBMO TST 2.3

In the rectangle $ABCD$ with $AB> BC$, the perpendicular bisecotr of $AC$ intersects the side $CD$ at point $E$. The circle with the center at point $E$ and the radius $AE$ intersects again the side $AB$ at point $F$. If point $O$ is the orthogonal projection of point $C$ on the line $EF$, prove that points $B, O$ and $D$ are collinear.

2012 Moldova JBMO TST 1.3

Let $ABC$ be an equilateral triangle, take line $t$ such that $t\parallel BC$ and $t$ passes through $A$. Let point $D$ be on side $AC$ , the bisector of angle $ABD$ intersects line $t$ in point $E$. Prove that $BD = CD + AE$.

2012 Moldova JBMO TST 2.3

Let $ABC$ be an isosceles triangle with $AC=BC$ . Take points $D$ on side $AC$ and $E$ on side $BC$ and $F$ the intersection of bisectors of angles $DEB$ and $ADE$ such that $F$ lies on side $AB$. Prove that $F$ is the midpoint of $AB$.

2013 Moldova JBMO TST 1.3
The point $O$ is the center of the circle circumscribed of the acute triangle $ABC$, and $H$ is the point of intersection of the heights of this triangle. Let $A_1, B_1, C_1$ be the points diametrically opposed to the vertices $A, B , C$  respectively  of the triangle, and $A_2, B_2, C_2$ be the midpoints of the segments $[AH], [BH] ¸[CH]$ respectively . Prove that the lines $A_1A_2, B_1B_2, C_1C_2$ are concurrent .

2013 Moldova JBMO TST 2.3
The points $M$ and $N$ are located respectively on the diagonal $(AC)$ and the side $(BC)$ of the square $ABCD$ such that $MN = MD$. Determine the measure of the angle $MDN$.

2014 Moldova JBMO TST 1.3
Let $ABC$ be a right triangle with $\angle ABC = 90^o$ . Points $D$ and $E$, located on the legs $(AC)$ and $(AB)$ respectively, are the legs of the inner bisectors taken from the vertices $B$ and $C$, respectively. Let $I$ be the center of the circle inscribed in the triangle $ABC$. If $BD \cdot CE = a^2 \sqrt2$ , find the area of the triangle $BIC$

2014 Moldova JBMO TST 2.3

Let the isosceles right triangle $ABC$ with  $\angle A = 90^o$ . The points  $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE   = 15^o$ and $CE = CF$. Determine the measure of the angle $CBF$.

2015 Moldova JBMO TST 1.3
Let $\Omega$ be the circle circumscribed to the triangle $ABC$. Tangents taken to the circle $\Omega$ at points $A$ and $B$ intersects at the point $P$ , and the perpendicular bisector of $(BC)$ cuts line  $AC$ at point $Q$. Prove that lines  $BC$ and $PQ$ are parallel.

2015 Moldova JBMO TST 2.3
In a right triangle $ABC$ with  $\angle BAC =90^o$and  $\angle ABC= 54^o$,  point $M$ is the midpoint of the hypotenuse $[BC]$ , point $D$ is the base of the bisector drawn from the vertex $C$ and $AM \cap CD = \{E\}$. Prove that $AB= CE$.

2016 Moldova JBMO TST 1.3
Let $ABC$ be an isosceles triangle with $\measuredangle C=\measuredangle B=36$. The point $M$ is in interior of $ABC$ such that $\measuredangle MBC=24^{\circ} , \angle BCM=30^{\circ}$ $N = AM \cap BC.$. Find $\angle ANC$ .

2016 Moldova JBMO TST 2.3

Let $ABCD$ ba a square and  let point $E$ be the midpoint of side $AD$. Points $G$ and $F$ are located on the segment $(BE)$  such that the lines $AG$ and $CF$ are perpendicular on the line $BE$. Prove that $DF= CG$.

2017 Moldova JBMO TST 1.3
Let $ABC$ be a triangle inscribed in a semicircle with center $O$ and diameter $BC$ .Two tangent lines to the semicircle at $A$ and $B$ intersect at $D.$ Prove that $DC$ goes through the midpoint of the altitude $AH$ of triangle $ABC.$

2017 Moldova JBMO TST 2.3
Given is an acute triangle $ABC$ and the median $AM.$ Draw $BH\perp AC$. The line which goes through $A$ and is perpendicular to $AM$ intersects $BH$ at $E.$ On the opposite ray of the ray $AE$ choose $F$ such that $AE=AF.$ Prove that $CF\perp AB$.

2018 Moldova JBMO TST 1.2
Let $ABC$ be an acute triangle.Let $OF \| BC$ where $O$ is the circumcenter and $F$ is between $A$ and $B$.Let $H$ be the orthocenter. Let $M$ be the midpoint of $AH$.Prove that $\angle FMC=90$.

2018 Moldova JBMO TST 2.3
Let $ABCD$ be a convex quadrilateral and $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$,and $O$ their intersection point.Point $M$ is the midpoint of $AB$ and $N$ is the midpoint of $CD$ such that $MN \cap AC ={E},MN \cap BD={F}$.Prove that $OE \cdot QF= OF\cdot PE$

2019 Moldova JBMO TST 1.3
Let $O$ be the center of circumscribed circle $\Omega$ of acute triangle $\Delta ABC$. The line $AC$ intersects the circumscribed circle of triangle $\Delta ABO$ for the second time in $X$. Prove that $BC$ and $XO$ are perpendicular.

2019 Moldova JBMO TST 2.3
Point $H$ is the orthocenter of the acute triangle $\Delta ABC$ and point $K$,situated on the line $(BC)$, is the foot of the perpendicular from point $A$ .The circle $\Omega$ passes through points $A$ and $K$ ,intersecting the sides $(AB)$ and $(AC)$ in points $M$ and $N$ .The line that passes through point $A$ and is parallel with $BC$ intersects again the circumcircles of triangles $\Delta AHM$ and $\Delta AHN$ in points $X$ and $Y$.Prove that $XY =BC$.

2019 Moldova JBMO TST 3.3
Let $I$ be the center of inscribed circle of right triangle $\Delta ABC$ with $\angle A = 90$ and point $M$ is the midpoint of $(BC)$.The bisector of $\angle BAC$ intersects the circumcircle of $\Delta ABC$ in point $W$.Point $U$ is situated on the line $AB$ such that the lines $AB$ and $WU$ are perpendiculars.Point $P$ is situated on the line $WU$ such that the lines $PI$ and $WU$ are perpendiculars.Prove that the line $MP$ bisects the segment $CI$.

2020 Moldova JBMO TST  p1

Let there be a triangle $ABC$ with orthocenter $H$. Let the lengths of the heights be $h_a, h_b, h_c$ from points $A, B$ and respectively $C$, and the semi-perimeter $p$ of triangle $ABC$.

It is known that $AH \cdot h_a + BH \cdot h_b + CH \cdot h_c = \frac{2}{3} \cdot p^2$. Show that $ABC$ is equilateral.

2020 Moldova JBMO TST  p6

The inscribed circle inside triangle $ABC$ intersects side $AB$ in $D$. The inscribed circle inside triangle $ADC$ intersects sides $AD$ in $P$ and $AC$ in $Q$.The inscribed circle inside triangle $BDC$ intersects sides $BC$ in $M$ and $BD$ in $N$. Prove that $P , Q, M, N$ are cyclic.

2020 Moldova JBMO TST  p11

Let $\triangle ABC$ be an acute triangle. The bisector of $\angle ACB$ intersects side $AB$ in $D$. The circumcircle of triangle $ADC$ intersects side $BC$ in $C$ and $E$ with $C \neq E$. The line parallel to $AE$ which passes through $B$ intersects line $CD$ in $F$. Prove that the triangle $\triangle  AFB$ is isosceles.

2021 Moldova  JBMO TST p2

Inside the parallelogram $ABCD$, point $E$ is chosen, such that $AE = DE$ and $\angle ABE = 90^o$. Point $F$ is the midpoint of the side $BC$ . Find the measure of the angle $\angle DFE$.

2021 Moldova  JBMO TST p5

Let $ABC$ be the triangle with $\angle ABC = 76^o$ and $\angle ACB  = 72^o$. Points $P$ and $Q$ lie on the sides $(AB)$ and $(AC)$, respectively, such that $\angle ABQ  = 22^o$ and $\angle ACP = 44^o$. Find the measure of angle $\angle APQ$.

2022 Moldova  JBMO TST p3

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A straight line is drawn through point $B$, which again intersects circles $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Point $E$, located on circle $\omega_1$ , satisfies the relation $CE = CB$ , and point $F$, located on circle $\omega_2$, satisfies the relation $DB = DF$. The line $BF$ intersects again the circle $\omega_1$ at the point $P$, and the line $BE$ intersects again the circle $\omega_2$ at the point $Q$. Prove that the points $A, P$, and $Q$ are collinear.

2022 Moldova  JBMO TST p8

Let $ABC$ be the triangle and $I$ the center of the circle inscribed in this triangle. The point $M$, located on the tangent taken to the point $B$ to the circumscribed circle of the triangle $ABC$, satisfies the relation $AB = MB$. Point $N$, located on the tangent taken to point $C$ to the same circle, satisfies the relation $AC = NC$. Points $M, A$ and $N$ lie on the same side of the line $BC$. Prove that $$\angle BAC + \angle MIN = 180^o.$$

2022 Moldova  JBMO TST p9

The circle inscribed in the triangle $ABC$ with center $I$ touches the side $BC$ at the point $D$. The line $DI$ intersects the side $AC$ at the point $M$. The tangent from $M$ to the inscribed circle, different from $AC$, intersects the side $AB$ at the point $N$. The line $NI$ intersects the side $BC$ at the point $P$. Prove that $AB = BP$.

source: https://pregatirematematicaolimpiadejuniori.wordpress.com/