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Romania JBMO TST 1998 - 2019, 2021-22 131p (+ ROMOP '11)

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

[not in JBMO Shortlist]

JBMO TST geometry collected inside aops  before 2022 here, after 2022 here
ROMOP 2011 Juniors inside aops here


Romania JBMO 1998 - 2019, 2021-22
1998 Romania JBMO TST 1.2
Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that:
a) $ p_{MNPQ}\ge AC+BD. $
b) $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $
c) $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $
Dan Brânzei and Gheorghe Iurea
1998 Romania JBMO TST 2.2
We´re given an inscriptible quadrilateral $ DEFG $ having some vertices on the sides of a triangle $ ABC, $ and some vertices (at least one of them) coinciding with the vertices of the same triangle. Knowing that the lines $ DF $ and $ EG $ aren´t parallel, find the locus of their intersection.

Dan Brânzei
1999 Romania JBMO TST 1.2
Consider, on a plane, the triangle $ ABC, $ vectors $ \vec x,\vec y,\vec z, $ real variable $ \lambda >0 $ and $ M,N,P $ such that
$ \left\{\begin{matrix} \overrightarrow{AM}=\lambda\cdot\vec x\\\overrightarrow{AN}=\lambda\cdot\vec y \\\overrightarrow{AP}=\lambda\cdot\vec z \end{matrix}\right. . $ Find the locus of the center of mass of $ MNP. $

Dan Brânzei and Gheorghe Iurea
1999 Romania JBMO TST 2.1
Find a relation between the angles of a triangle such that this could be separated in two isosceles triangles by a line.
Dan Brânzei
1999 Romania JBMO TST 2.4
Let be a convex quadrilateral $ ABCD. $ On the semi-straight line extension of $ AB $ in the direction of $ B, $ put $ A_1 $ such that $ AB=BA_1. $ Similarly, define $ B_1,C_1,D_1, $ for the other three sides.
a) If $ E,E_1,F,F_1 $ are the midpoints of $ BC,A_1B_1,AD $ respectively, $ C_1,D_1, $ show that $ EE_1=FF_1. $
b) Delete everything, but $ A_1,B_1,C_1,D_1. $ Now, find a way to construct the initial quadrilateral.

On the hypotenuse $ BC $ of an isosceles right triangle $ ABC $ let $ M,N $ such that $ BM^2-MN^2+NC^2=0. $ Show that $ \angle MAN= 45^{\circ } . $

Let be a triangle $ ABC, $ and three points $ A',B',C' $ on the segments $ BC,CA, $ respectively, $ AB, $ such that the lines $ AA',BB',CC' $ are concurent at $ M. $ Name $ a,b,c,x,y,z $ the areas of the triangles $ AB'M,BC'M,CA'M,AC'M,BA'M, $ respectively, $ CB'M. $ Show that:
a) $ abc=xyz $
b) $ ab+bc+ca=xy+yz+zx $

Let $ D,E,F $ be the feet of the interior bisectors from $ A,B, $ respectively $ C, $ and let $ A',B',C' $ be the symmetric points of $ A,B, $ respectively, $ C, $ to $ D,E, $ respectively $ F, $ such that $ A,B,C $ lie on $ B'C',A'C', $ respectively, $ A'B'. $ Show that the $ ABC $ is equilateral.

Let $ABC$ be an arbitrary triangle. A circle passes through $B$ and $C$ and intersects the lines $AB$ and $AC$ at $D$ and $E$, respectively. The projections of the points $B$ and $E$ on $CD$ are denoted by $B'$ and $E'$, respectively. The projections of the points $D$ and $C$ on $BE$ are denoted by $D'$ and $C'$, respectively. Prove that the points $B',D',E'$ and $C'$ lie on the same circle.

2001 Romania JBMO TST 2.1
Let $ABCD$ be a rectangle. We consider the points $E\in CA,F\in AB,G\in BC$ such that $DC\perp CA,EF\perp AB$ and $EG\perp BC$. Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$.

2001 Romania JBMO TST 2.3
Let $ABCD$ be a quadrilateral inscribed in the circle $O$. For a point $E\in O$, its projections $K,L,M,N$ on the lines $DA,AB,BC,CD$, respectively, are considered. Prove that if $N$ is the orthocentre of the triangle $KLM$ for some point $E$, different from $A,B,C,D$, then this holds for every point $E$ of the circle.

2001 Romania JBMO TST 3.2
Let $ABCDEF$ be a hexagon with $AB||DE,\ BC||EF,\ CD||FA$ and in which the diagonals $AD,BE$ and $CF$ are congruent. Prove that the hexagon can be inscribed in a circle.

2001 Romania JBMO TST 3.4
Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.

2002 Romania JBMO TST 1.4
Let $ABCD$ be a parallelogram of center $O$. Points $M$ and $N$ are the midpoints of $BO$ and $CD$, respectively. Prove that if the triangles $ABC$ and $AMN$ are similar, then $ABCD$ is a square.

2002 Romania JBMO TST 2.3
Let $ABC$ be an isosceles triangle such that $AB = AC$ and $\angle A = 20^o$. Let $M$ be the foot of the altitude from C and let $N$ be a point on the side $AC$ such that $CN =\frac12 BC$.  Determine the measure of the angle $AMN$.

2002 Romania JBMO TST 2.4 (also IMO TST 2.1)
Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$.

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ meet at $O$. Let $m$ be the measure of the acute angle formed by these diagonals. A variable angle $xOy$ of measure $m$ intersects the quadrilateral by a convex quadrilateral of constant area. Prove that $ABCD$ is a square.

Let $C_1(O_1)$ and $ C_2(O_2)$ be two circles such that $C_1$ passes through $O_2$. Point $M$ lies on $C_1$ such that $M \notin O_1O_2$. The tangents from $M$ at $O_2$ meet again $C_1$ at $A$ and $B$. Prove that the tangents from $A$ and $B$ at $C_2$ - others than $MA$ and $MB$ - meet at a point located on $C_1$.

We are given $n$ circles which have the same center. Two lines $D_1,D_2$ are concurent in $P$, a point inside all circles. The rays determined by $P$ on the line $D_i$ meet the circles in points $A_1,A_2,...,A_n$ and $A'_1, A'_2,..., A'_n$ respectively and the rays on $D_2$ meet the circles at points $B_1,B_2, ... ,B_n$ and $B'_2, B'_2 ..., B'_n$ (points with the same indices lie on the same circle). Prove that if the arcs $A_1B_1$ and $A_2B_2$ are equal then the arcs $A_iB_i$ and $A'_iB'_i$ are equal, for all $i = 1,2,... n$.

Let $ABC$ be a triangle and $a = BC, b = CA$ and $c = AB$ be the lengths of its sides. Points $D$ and $E$  lie in the same halfplane determined by $BC$ as $A$. Suppose that $DB = c, CE = b$ and that the area of $DECB$ is maximal. Let $F$ be the midpoint of $DE$ and let $FB = x$. Prove that $FC = x$ and $4x^3 = (a^2+b^2 + c^2)x + abc$.

Consider a rhombus $ABCD$ with center $O$. A point $P$ is given inside the rhombus, but not situated on the diagonals. Let $M,N,Q,R$ be the projections of $P$ on the sides $(AB), (BC), (CD), (DA)$, respectively. The perpendicular bisectors of the segments $MN$ and $RQ$ meet at $S$ and the perpendicular bisectors of the segments $NQ$ and $MR$ meet at $T$. Prove that $P, S, T$ and $O$ are the vertices of a rectangle.

Two circles $C_1(O_1)$ and $C_2(O_2)$  with distinct radii meet at points $A$ and $B$. The tangent from $A$ to $C_1$ intersects the tangent from $B$ to $C_2$ at point $M$. Show that both circles are seen from $M$ under the same angle.

Let $E$ be the midpoint of the side $CD$ of a square $ABCD$. Consider the point $M$ inside the square such that  $\angle MAB = \angle MBC = \angle BME = x$. Find the angle $x$.

Suppose $ABCD$ and $AEFG$ are rectangles such that the points $B,E,D,G$ are collinear (in this order). Let the lines $BC$ and $GF$ intersect at point $T$ and let the lines $DC$ and $EF$ intersect at point $H$. Prove that points $A, H$ and $T$ are collinear.

2004 Romania JBMO TST 1.3
Let $V$ be a point in the exterior of a circle of center $O$, and let $T_1,T_2$ be the points where the tangents from $V$ touch the circle. Let $T$ be an arbitrary point on the small arc $T_1T_2$. The tangent in $T$ at the circle intersects the line $VT_1$ in $A$, and the lines $TT_1$ and $VT_2$ intersect in $B$. We denote by $M$ the intersection of the lines $TT_1$ and $AT_2$.
Prove that the lines $OM$ and $AB$ are perpendicular.

Let $ABC$ be a triangle, having no right angles, and let $D$ be a point on the side $BC$. Let $E$ and $F$ be the feet of the perpendiculars drawn from the point $D$ to the lines $AB$ and $AC$ respectively. Let $P$ be the point of intersection of the lines $BF$ and $CE$. Prove that the line $AP$ is the altitude of the triangle $ABC$ from the vertex $A$ if and only if the line $AD$ is the angle bisector of the angle $CAB$.

Let $ABC$ be an isosceles triangle with $AB=AC$. Consider a variable point $P$ on the extension of the segment $BC$ beyound $B$ (in other words, $P$ lies on the line $BC$ such that the point $B$ lies inside the segment $PC$). Let $r_{1}$ be the radius of the incircle of the triangle $APB$, and let $r_{2}$ be the radius of the $P$-excircle of the triangle $APC$. Prove that the sum $r_{1}+r_{2}$ of these two radii remains constant when the point $P$ varies.
Virgil Nicula
Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.

Let $ABC$ be a triangle inscribed in the circle $K$ and consider a point $M$ on the arc $BC$ that do not contain $A$. The tangents from $M$ to the incircle of $ABC$ intersect the circle $K$ at the points $N$ and $P$. Prove that if $\angle BAC = \angle NMP$, then triangles $ABC$ and $MNP$ are congruent.
Valentin Vornicu


Let $M,N, P$ be the midpoints of the sides $BC,CA,AB$ of the triangle $ABC$, respectively, and let $G$ be the centroid of the triangle. Prove that if $BMGP$ is cyclic and $2BN = \sqrt3 AB$ , then triangle $ABC$ is equilateral.

2005 Romania JBMO TST 1.1
Let $\mathcal{C}_1(O_1)$ and $\mathcal{C}_2(O_2)$ be two circles which intersect in the points $A$ and $B$. The tangent in $A$ at $\mathcal{C}_2$ intersects the circle $\mathcal{C}_1$ in $C$, and the tangent in $A$ at $\mathcal{C}_1$ intersects $\mathcal{C}_2$ in $D$. A ray starting from $A$ and lying inside the $\angle CAD$ intersects the circles $\mathcal{C}_1$, $\mathcal{C}_2$ in the points $M$ and $N$ respectively, and the circumcircle of $\triangle ACD$ in $P$. Prove that $AM=NP$.

 2005 Romania JBMO TST 2.2
On the sides $AD$ and $BC$ of a rhombus $ABCD$ we consider the points $M$ and $N$ respectively. The line $MC$ intersects the segment $BD$ in the point $T$, and the line $MN$ intersects the segment $BD$ in the point $U$. We denote by $Q$ the intersection between the line $CU$ and the side $AB$ and with $P$ the intersection point between the line $QT$ and the side $CD$. Prove that the triangles $QCP$ and $MCN$ have the same area.

2005 Romania JBMO TST 2.3
 Let $ABC$ be an equilateral triangle and $M$ be a point inside the triangle. We denote by $A'$, $B'$, $C'$ the projections of the point $M$ on the sides $BC$, $CA$ and $AB$ respectively. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent if and only if $M$ belongs to an altitude of the triangle.

2005 Romania JBMO TST 3.3
Let $ABC$ be a triangle with $BC>CA>AB$ and let $G$ be the centroid of the triangle. Prove that $ \angle GCA+\angle GBC<\angle BAC<\angle GAC+\angle GBA . $
Three circles $\mathcal C_1(O_1)$, $\mathcal C_2(O_2)$ and $\mathcal C_3(O_3)$ share a common point and meet again pairwise at the points $A$, $B$ and $C$. Show that if the points $A$, $B$, $C$ are collinear then the points $Q$, $O_1$, $O_2$ and $O_3$ lie on the same circle.

2005 Romania JBMO TST 5.2
Let $AB$ and $BC$ be two consecutive sides of a regular polygon with 9 vertices inscribed in a circle of center $O$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of the radius perpendicular to $BC$. Find the measure of the angle $\angle OMN$

2005 Romania JBMO TST 5.3
A piece of cardboard has the shape of a pentagon $ABCDE$ in which $BCDE$ is a square and $ABE$ is an isosceles triangle with a right angle at $A$. Prove that the pentagon can be divided in two different ways in three parts that can be rearranged in order to recompose a right isosceles triangle.

2006 Romania JBMO TST 1.1
Let $ABC$ be a triangle right in $C$ and the points $D, E$ on the sides $BC$ and $CA$ respectively, such that $\frac{BD}{AC} =\frac{AE}{CD} = k$. Lines $BE$ and $AD$ intersect at $O$. Show that the angle  $\angle BOD = 60^o$  if  and only if $k =\sqrt3$.

2006 Romania JBMO TST 2.2
Let $C (O)$ be a circle (with center $O$ )  and $A, B$ points on the circle with $\angle AOB = 90^o$. Circles $C_1 (O_1)$ and $C_2 (O_2)$ are tangent internally with circle $C$ at $A$ and $B$, respectively, and, also, are tangent to each other. Consider another circle $C_3 (O_3)$ tangent externally to the circles $C_1, C_2$ and tangent internally to circle $C$, located inside angle $\angle AOB$. Show that the points $O, O_1, O_2, O_3$ are the vertices of a rectangle.

2006 Romania JBMO TST 3.1
Let $ABCD$ be a cyclic quadrilateral of area 8. If there exists a point $O$ in the plane of the quadrilateral such that $OA+OB+OC+OD = 8$, prove that $ABCD$ is an isosceles trapezoid.

2006 Romania JBMO TST 4.2
Let $ABC$ be a triangle and $A_1$, $B_1$, $C_1$ the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Prove that if $M$ is a point in the plane of the triangle such that $ \frac{MA}{MA_1} = \frac{MB}{MB_1} = \frac{MC}{MC_1} = 2 , $ then $M$ is the centroid of the triangle.

2006 Romania JBMO TST 5.1
Let $ABC$ be a triangle and $D$ a point inside the triangle, located on the median of $A$. Prove that if $\angle BDC = 180^o - \angle BAC$, then $AB \cdot CD = AC \cdot  BD$.

2007 Romania JBMO TST 1.2
Let $ABCD$ be a trapezium $(AB \parallel CD)$ and $M,N$ be the intersection points of the circles of diameters $AD$ and $BC$. Prove that $O \in MN$, where $O \in AC \cap BD$.

2007 Romania JBMO TST 2.2
Consider a convex quadrilateral $ABCD$. Denote $M, \ N$ the points of tangency of the circle inscribed in $\triangle ABD$ with $AB, \ AD$, respectively and $P, \ Q$ the points of tangency of the circle inscribed in $\triangle CBD$ with the sides $CD, \ CB$, respectively. Assume that the circles inscribed in $\triangle ABD, \ \triangle CBD$ are tangent. Prove that:
a) $ABCD$ is circumscriptible.
b) $MNPQ$ is cyclic.
c) The circles inscribed in $\triangle ABC, \ \triangle ADC$ are tangent.

2007 Romania JBMO TST 2.3
Let $ABC$ an isosceles triangle, $P$ a point belonging to its interior. Denote $M$, $N$ the intersection points of the circle $\mathcal{C}(A, AP)$ with the sides $AB$ and $AC$, respectively.
Find the position of $P$ if $MN+BP+CP$ is minimum.

2007 Romania JBMO TST 3.1
Let $ABC$ a triangle and $M,N,P$ points on $AB,BC$, respective $CA$, such that the quadrilateral $CPMN$ is a paralelogram. Denote $R \in AN \cap MP$, $S \in BP \cap MN$, and $Q \in AN \cap BP$. Prove that $[MRQS]=[NQP]$.

2007 Romania JBMO TST 4.2
Let $w_{1}$ and $w_{2}$ be two circles which intersect at points $A$ and $B$. Consider $w_{3}$ another circle which cuts $w_{1}$ in $D,E$, and it is tangent to $w_{2}$ in the point $C$, and also tangent to $AB$ in $F$. Consider $G \in DE \cap AB$, and $H$ the symetric point of $F$ w.r.t $G$. Find $\angle{HCF}$.

2007 Romania JBMO TST 5.1
Consider $ \rho$ a semicircle of diameter $ AB$. A parallel to $ AB$ cuts the semicircle at $ C, D$ such that $ AD$ separates $ B, C$. The parallel at $ AD$ through $ C$ intersects the semicircle the second time at $ E$. Let $ F$ be the intersection point of the lines $ BE$ and $ CD$. The parallel through $ F$ at $ AD$ cuts $ AB$ in $ P$. Prove that $ PC$ is tangent to $ \rho$.
Cosmin Pohoata
2007 Romania JBMO TST 6.3
Let $ABC$ be a right triangle with $A = 90^{\circ}$ and $D \in (AC)$. Denote by $E$ the reflection of $A$ in the line $BD$ and $F$ the intersection point of $CE$ with the perpendicular in $D$ to $BC$. Prove that $AF, DE$ and $BC$ are concurrent.

2008 Romania JBMO TST 1.3
Let $ ABC$ be an acute-angled triangle. We consider the equilateral triangle $ A'UV$, where $ A' \in (BC)$, $ U\in (AC)$ and $ V\in(AB)$ such that $ UV \parallel BC$. We define the points $ B',C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.

2008 Romania JBMO TST 1.4
Let $ ABC$ be a triangle, and $ D$ the midpoint of the side $ BC$. On the sides $ AB$ and $ AC$ we consider the points $ M$ and $ N$, respectively, both different from the midpoints of the sides, such that $AM^2+AN^2 =BM^2 + CN^2 $  and $\angle MDN = \angle BAC$. Prove that $ \angle BAC = 90^\circ$.

2008 Romania JBMO TST 2.1
Consider the acute-angled triangle $ ABC$, altitude $ AD$ and point $ E$ - intersection of $ BC$ with diameter from $ A$ of circumcircle. Let $ M,N$ be symmetric points of $ D$ with respect to the lines $ AC$ and $ AB$ respectively. Prove that $ \angle{EMC} = \angle{BNE}$.

2008 Romania JBMO TST 3.4
Let $ d$ be a line and points $ M,N$ on the $ d$. Circles $ \alpha,\beta,\gamma,\delta$ with centers $ A,B,C,D$ are tangent to $ d$, circles $ \alpha,\beta$ are externally tangent at $ M$, and circles $ \gamma,\delta$ are externally tangent at $ N$. Points $ A,C$ are situated in the same half-plane, determined by $ d$. Prove that if exists an circle, which is tangent to the circles $ \alpha,\beta,\gamma,\delta$ and contains them in its interior, then lines $ AC,BD,MN$ are concurrent or parallel.

2008 Romania JBMO TST 4.1
Let $ ABCD$ be a convex quadrilateral with opposite side not parallel. The line through $ A$ parallel to $ BD$ intersect line $ CD$ in $ F$, but parallel through $ D$ to $ AC$ intersect line $ AB$ at $ E$. Denote by $ M,N,P,Q$ midpoints of the segments $ AC,BD,AF,DE$. Prove that lines $ MN,PQ$ and $ AD$ are concurrent.


2009 Romania JBMO TST 1.2
Consider a rhombus $ABCD$. Point $M$ and $N$ are given on the line segments $AC$ and $BC$ respectively, such that $DM = MN$. Lines $AC$ and $DN$ meet at point $P$ and lines $AB$ and $DM$ meet at point $R$. Prove that $RP = PD$.
2009 Romania JBMO TST 2.2
Let $ABCD$ be a quadrilateral. The diagonals $AC$ and $BD$ are perpendicular at point $O$. The perpendiculars from $O$ on the sides of the quadrilateral meet $AB, BC, CD, DA$ at $M, N, P, Q$, respectively, and meet again $CD, DA, AB, BC$ at $M', N', P', Q'$, respectively. Prove that points $M, N, P, Q, M', N', P', Q'$ are concyclic.
Cosmin Pohoata
2009 Romania JBMO TST 2.3
Consider a regular polygon $A_0A_1...A_{n-1}, n \ge 3$, and $m \in\{1, 2, ..., n - 1\}, m \ne n/2$. For any number $i \in \{0,1, ... , n - 1\}$, let $r(i)$ be the remainder of $i + m$ at the division by $n$. Prove that no three segments $A_iA_{r(i)}$ are concurrent.

2009 Romania JBMO TST 3.4
Consider $K$ a polygon in plane, such that the distance between any two vertices is not greater than $1$. Let $X$ and $Y$ be two points inside $K$. Show that there exist a point $Z$, lying on the border of K, such  that $XZ + Y Z \le 1$

 2009 Romania JBMO TST 4.1
Show that in any triangle $ABC$ with $A = 90^0$ the following inequality holds:
$$(AB -AC)^2(BC^2 + 4AB \cdot  AC)^2 \le 2BC^6$$

2009 Romania JBMO TST 4.3
Let $ABC$ be a triangle and $A_1$ the foot of the internal bisector of angle $BAC$. Consider $d_A$ the perpendicular line from $A_1$ on $BC$. Define analogously the lines $d_B$ and $d_C$. Prove that lines $d_A, d_B$ and $d_C$ are concurrent if and only if triangle $ABC$ is isosceles.

2010 Romania JBMO TST 1.2
Let $ABCD$ be a convex quadrilateral with $\angle BCD= 120^o, \angle {CBA} = 45^o, \angle {CBD} = 15^o$ and $\angle {CAB} = 90^o$. Show that $AB = AD$.

 2010 Romania JBMO TST 1.4
Let $ABC$ be an isosceles triangle with $AB = AC$ and let $n$  be a natural number, $n>1$. On the side $AB$ we consider the point $M$ such that $n \cdot AM = AB$. On the side $BC$ we consider the points $P_1, P_2, ....., P_ {n-1}$ such that $BP_1 = P_1P_2 = .... = P_ {n-1} C = \frac{1}{n} BC$.

Show that: $\angle {MP_1A} + \angle {MP_2A} + .... + \angle {MP_ {n-1} A} = \frac{1} {2} \angle {BAC}$.


2010 Romania JBMO TST 2.4
Let $I$ be the incenter of scalene triangle ABC and denote by $a,$ $b$ the circles with diameters $IC$ and $IB$, respectively. If $c,$ $d$ mirror images of $a,$ $b$ in $IC$ and $IB$ prove that the circumcenter $O$ of triangle $ABC$ lies on the radical axis of $c$ and $d$.
Cosmin Pohoata
2010 Romania JBMO TST 3.1
Consider two equilateral triangles $ABC$ and $MNP$ with the property that $AB \parallel MN, BC \parallel NP$ and $CA \parallel PM$ , so that the surfaces of the triangles intersect after a convex hexagon. The distances between the three pairs of parallel lines are at most equal to $1$. Show that at least one of the two triangles has the side at most equal to $\sqrt {3}$ 

2010 Romania JBMO TST 3.4
Let a triangle  $ABC$ , $O$ it's circumcenter , $H$ ortocenter and $M$  the midpoint of  $AH$. The perpendicular at  $M$  to line $OM$ meets  $AB$  and  $AC$ at points  $P$, respective  $Q$. Prove that $MP=MQ$.
Cosmin Pohoata
2010 Romania JBMO TST 4.3
Let $ABC$ be a triangle inscribed in the circle $(O)$. Let $I$ be the center of the circle inscribed in the triangle and $D$  the point of contact of the circle inscribed with the side $BC$. Let $M$ be the second point of intersection of the bisector $AI$ with the circle $(O)$ and let $P$ be the point where the line $DM$ intersects the circle $(O)$ . Show that $PA \perp PI$.

2010 Romania JBMO TST 5.2
Let $ABC$ be a triangle and $D, E, F$ the midpoints of the sides $BC, CA, AB$ respectively. Show that $\angle DAC = \angle ABE$ if and only if $\angle AFC = \angle BDA$


Let $A_1A_2A_3A_4A_5$ be a convex pentagon. Suppose rays $A_2A_3$ and $A_5A_4$ meet at the point $X_1$. Define $X_2$, $X_3$, $X_4$, $X_5$ similarly. Prove that $$\displaystyle\prod_{i=1}^{5} X_iA_{i+2} = \displaystyle\prod_{i=1}^{5} X_iA_{i+3}$$
where the indices are taken modulo 5.

2011 Romania JBMO TST 2.4
The measure of the angle $\angle A$ of the acute triangle $ABC$ is $60^o$, and $HI = HB$, where $I$ and $H$ are the incenter and the orthocenter of the triangle $ABC$. Find the measure of the angle  $\angle  B$.

2011 Romania JBMO TST 3.3
Let $ABC$ be a triangle, $I_a$ the center of the excircle at side $BC$, and $M$ its reflection across $BC$. Prove that $AM$ is parallel to the Euler line of the triangle $BCI_a$.

2011 Romania JBMO TST 4.2 (IMO 2009)
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP = OQ.$

Sergei Berlov, Russia
Consider the triangle $ABC$ and the points $D \in (BC)$ and $M \in (AD)$. Lines $BM$ and $AC$ meet at $E$, lines $CM$ and $AB$ meet at $F$, and lines $EF$ and $AD$ meet at $N$. Prove that $\frac{AN}{DN}=\frac{1}{2}\cdot \frac{AM}{DM}$

Let $ABC$ be a triangle and $A', B', C'$ the points in which its incircle touches the sides $BC, CA, AB$, respectively. We denote by $I$ the incenter and by $P$ its projection onto $AA' $. Let $M$ be the midpoint of the line segment $[A'B']$ and $N$ be the intersection point of the lines $MP$ and $AC$. Prove that $A'N $is parallel to $B'C'$

The quadrilateral $ABCD$ is inscribed in a circle centered at $O$, and $\{P\} = AC \cap BD, \{Q\} = AB \cap CD$. Let $R$ be the second intersection point of the circumcircles of the triangles $ABP$ and $CDP$.
a) Prove that the points $P, Q$, and $R$ are collinear.
b) If $U$ and $V$ are the circumcenters of the triangles $ABP$, and $CDP$, respectively, prove that the points $U, R, O, V$ are concyclic.

Consider a semicircle of center $O$ and diameter $[AB]$, and let $C$ be an arbitrary point on the segment $(OB)$. The perpendicular to the line $AB$ through $C$ intersects the semicircle in $D$. A circle centered in $P$ is tangent to the arc $BD$ in $F$ and to the segments $[AB]$ and $[CD]$ in $G$ and $E$, respectively. Prove that the triangle $ADG$ is isosceles.

2012 Romania JBMO TST 4.3 (also) (generalisation of MOP 1997) 
Let $ABC$ be an arbitrary triangle, and let $M, N, P$ be any three points on the sides $BC, CA, AB$ such that the lines $AM, BN, CP$ concur. Let the parallel to the line $AB$ through the point $N$ meet the line $MP$ at a point $E$, and let the parallel to the line $AB$ through the point $M$ meet the line $NP$ at a point $F$. Then, the lines $CP, MN$ and $EF$ are concurrent.

2013 Romania JBMO TST 1.3
Consider a circle centered at $O$ with radius $r$ and a line $\ell$ not passing through $O$. A grasshopper is jumping to and fro between the points of the circle and the line, the length of each jump being $r$. Prove that there are at most $8$ points for the grasshopper to reach.

In the acute-angled triangle $ABC$, with $AB \ne AC$, $D$ is the foot of the angle bisector of angle $A$, and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. The circumcircles of triangles $DBF$ and $DCE$ intersect for the second time at $M$. Prove that $ME = MF$.

Leonard Giugiuc
Let $H$ be the orthocenter of an acute-angled triangle $ABC$ and $P$ a point on the circumcenter of triangle $ABC$. Prove that the Simson line of $P$ bisects the segment $[P H]$.

Let $ABCD$ be a cyclic quadrilateral and $\omega_1, \omega_2$ the incircles of triangles $ABC$ and $BCD$. Show that the common external tangent line of $\omega_1$ and $\omega_2$,  the other one than $BC$, is parallel with $AD$

2013 Romania JBMO TST 4.1
Let $A$ be a point on a semicircle of diameter $[BC]$, and $X$ an arbitrary point inside the triangle $ABC$. The line $BX$ intersects the semicircle for the second time in $K$, and intersects the line segment $(AC)$ in $F$. The line $CX$ intersects the semicircle for the second time in in $L$, and intersects the segment line $(AB)$ in $E$. Prove that the circumcircles of triangles $AKF$ and $AEL$ are tangent

2013 Romania JBMO TST 4.3
Let $D$ be the midpoint of the side $[BC]$ of the triangle $ABC$ with $AB \ne AC$ and $E$ the foot of the altitude from $BC$. If $P$ is the intersection point of the perpendicular bisector of the segment line $[DE]$ with the perpendicular from $D$ onto the the angle bisector of $BAC$, prove that $P$ is on the Euler circle of triangle $ABC$.

Consider acute triangles $ABC$ and $BCD$, with $\angle BAC = \angle BDC$, such that $A$ and $D$ are on opposite sides of line $BC$. Denote by $E$ the foot of the perpendicular line to $AC$ through $B$ and by $F$ the foot of the perpendicular line to $BD$ through $C$. Let $H_1$ be the orthocenter of triangle $ABC$ and $H_2$ be the orthocenter of $BCD$. Show that  lines $AD, EF$ and $H_1H_2$ are concurrent.
2014 Romania JBMO TST 1.4
Let $ABCD$ be a quadrilateral with $\angle A + \angle C = 60^o$. If $AB \cdot   CD = BC \cdot AD$, prove that $AB \cdot  CD = AC \cdot   BD$.

Leonard Giugiuc
2014 Romania JBMO TST 1.5
Let $D$ and $E$ be the midpoints of sides $[AB]$ and $[AC]$ of the triangle $ABC$. The circle of diameter $[AB]$ intersects the line $DE$ on the opposite side of $AB$ than $C$, in $X$. The circle of diameter $[AC]$ intersects $DE$ on the opposite side of $AC$ than $B$ in $Y$ . Prove that the orthocenter of triangle $XY T$ lies on $BC$.

2014 Romania JBMO TST 2.3
Let $ABC$ be an acute triangle and $D \in (BC) , E \in (AD)$ be mobile points. The circumcircle of triangle $CDE$ meets the median from $C$ of the triangle $ABC$ at $F$ Prove that the circumcenter of triangle $AEF$ lies on a fixed line.

2014 Romania JBMO TST 3.4
In the acute triangle $ABC$, with $AB \ne BC$, let $T$ denote the midpoint of the side $[AC], A_1$ and $C_1$ denote the feet of the altitudes drawn from $A$ and $C$, respectively. Let $Z$ be the point of intersection of the tangents in $A$ and $C $to the circumcircle of triangle $ABC, X$ be the point of intersection of lines $ZA$ and $A_1C_1$ and $Y$ be the point of intersection of lines $ZC$ and $A_1C_1$.
a) Prove that $T$ is the incircle of triangle $XYZ$.
b) The circumcircles of triangles $ABC$ and $A_1BC_1$ meet again at $D$. Prove that the orthocenter $H$ of triangle $ABC$ is on the line $TD$.
c) Prove that the point $D$ lies on the circumcircle of triangle $XYZ$.

2014 Romania JBMO TST 4.4
In a circle, consider two chords $[AB], [CD]$ that intersect at $E$, lines $AC$ and $BD$ meet at $F$. Let $G$ be the projection of $E$ onto $AC$. We denote by $M,N,K$ the midpoints of the segment lines $[EF] ,[EA]$ and $[AD]$, respectively. Prove that the points $M, N,K,G$ are concyclic.

2014 Romania JBMO TST 5.3
Let ABC be an acute triangle and let O be its circumcentre. Now, let the  diameter PQ of circle ABC intersects sides AB and AC in their interior at D and E, respectively. Now, let F and G be the midpoints of CD and BE. Prove that <FOG=<BAC

2015 Romania JBMO TST 1.1
Let $ABC$ be an acute triangle with $AB \neq AC$ . Also let $M$ be the midpoint of the side $BC$ , $H$ the orthocenter of the triangle $ABC$ , $O_1$ the midpoint of the segment $AH$ and $O_2$ the center of the circumscribed circle of the triangle $BCH$ . Prove that $O_1AMO_2$ is a parallelogram .

2015 Romania JBMO TST 1.5
Let $ABCD$ be a convex quadrilateral with non perpendicular diagonals and with the sides $AB$ and $CD$ non parallel . Denote by $O$ the intersection of the diagonals , $H_1$ the orthocenter of the triangle $AOB$ and $H_2$ the orthocenter of the triangle $COD$ . Also denote with $M$ the midpoint of the side $AB$ and with $N$ the midpoint of the side $CD$ . Prove that $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$

2015 Romania JBMO TST 2.4
Let $ABC$ be a triangle with $AB \neq BC$ and let $BD$ the interior bisectrix of  $ \angle ABC$ with $D \in AC$ . Let $M$ be the midpoint of the arc $AC$ that contains the point $B$ in the circumcircle of the triangle $ABC$ .The circumcircle of the triangle $BDM$ intersects the segment $AB$ in $K \neq B$ . Denote by $J$ the symmetric of $A$ with respect to $K$ .If $DJ$ intersects $AM$ in $O$ then prove that $J,B,M,O$ are concyclic.

2015 Romania JBMO TST 3.3
Let $ABC$ be an acute triangle , with $AB \neq AC$ and denote its orthocenter by $H$ . The point $D$ is located on the side $BC$ and the circumcircles of the triangles $ABD$ and $ACD$ intersects for the second time the lines $AC$ , respectively $AB$ in the points $E$ respectively $F$. If we denote by $P$ the intersection point of $BE$ and $CF$ then show that $HP \parallel BC$ if and only if $AD$ passes through the circumcenter of the triangle $ABC$.

2015 Romania JBMO TST 4.3
Let $ABC$ be a triangle with $AB \ne  AC$ and $ I$ its incenter. Let $M$ be the midpoint of the side $BC$ and $D$ the projection of $I$ on $BC.$ The line $AI$ intersects the circle with center $M$ and radius $MD$ at $P$ and $Q.$ Prove that $\angle BAC + \angle PMQ  = 180^{\circ}.$

Let $ABC$ be a triangle inscribed in circle $\omega$ and $P$ a point in its interior. The lines $AP,BP$ and $CP$ intersect circle $\omega$ for the second time at $D,E$ and $F,$ respectively. If $A',B',C'$ are the reflections of $A,B,C$ with respect to the lines $EF,FD,DE,$ respectively, prove that the triangles $ABC$ and $A'B'C'$ are similar.

Let $ABC$ be a acute triangle where $\angle BAC =60$. Prove that if the Euler's line of $ABC$ intersects $AB,AC$ in $D,E$, then $ADE$ is equilateral.

Let $ABC$ be an acute triangle with $AB<AC$ and $D,E,F$ be the contact points of the incircle $(I)$ with $BC,AC,AB$. Let $M,N$ be on $EF$ such that $MB \perp BC$ and $NC \perp BC$. $MD$ and $ND$ intersect the $(I)$ in $D$ and $Q$. Prove that $DP=DQ$.
Ruben Dario & Leo Giugiuc
2016 Romania JBMO TST 2.1
Triangle $\triangle{ABC},O$ is circumcenter of $(ABC), OA=R$, the $A$-excircle intersect $(AB),(BC),(CA)$ at points $F,D,E$. If the $A$-excircle has radius R prove that $OD\perp EF$.

2016 Romania JBMO TST 3.3
ABCD = cyclic quadrilateral , $AC\cap BD=X$
AA' $\perp $BD,A'$\in$ BD, CC' $\perp $BD,C'$\in$ BD
BB' $\perp $AC,B'$\in$ AC, DD' $\perp $AC,D'$\in$ AC . Prove that:
a) perpendiculars from midpoints of the sides to the opposite sides are concurrent.The point is called Mathot Point
b) A',B',C',D' are concyclic
c) if O'=circumcenter of (A'B'C') prove that O'=midpoint of the line that connects the orthocente of triangle XAB and XCD
d) O' is the Mathot Point

2016 Romania JBMO TST 4.1
The altitudes $AA_1$,$BB_1$,$CC_1$ of $\triangle{ABC}$ intersect at $H$.$O$ is the circumcenter of $\triangle{ABC}$.Let $A_2$ be the reflection of $A$ wrt $B_1C_1$.Prove that:
a) $O$,$A_2$,$B_1$,$C$ are all on a circle
b) $O$,$H$,$A_1$,$A_2$ are all on a circle

Let $ABCD$ be a cyclic quadrilateral.$E$ is the midpoint of $(AC)$ and $F$ is the midpoint of $(BD)$ {$G$}=$AB\cap CD$ and {$H$}=$AD\cap BC$.
a) Prove that the intersections of the angle bisector of $\angle{AHB}$ and the sides $AB$ and $CD$ and the intersections of the angle bisector of$\angle{AGD}$ with $BC$ and $AD$ are the verticles of a rhombus
b) Prove that the center of this rhombus lies on $EF$

2017 Romania JBMO TST 1.1
Let $P$ be a point in the interior of the acute-angled triangle $ABC$. Prove that if the reflections of $P$ with respect to the sides of the triangle lie on the circumcircle of the triangle, then $P$ is the orthocenter of $ABC$.

The incircle of triangle $ABC$ touches the sides $BC, CA$, and $AB$ at $D, E$, and $F$ respectively. On the line segments $EF, FD$, and $DE$, consider the points $M, N$, and $P$ respectively such that the sums $BM +MC, CN + NA$, and $AP + PB$ are minimum.
a) Prove that the lines $AM, BN$, and $CP$ are concurrent.
b) Prove that $DM, EN$ and $FP$ are the altitudes of triangle $DEF$.
Gabriel Popa
further remarks by Mircea Fianu
Triangles $ABC$ and $MNP$ are homothetic, the lines $AM, BN$ and $CN$ being concurrent in the center of homothety, $X$. Point $X$ lies on the Euler line of triangle $DEF$.  

Let $I$ be the incenter of the scalene $\Delta ABC$, such, $AB<AC$, and let $I'$ be the reflection of point $I$ in line $BC$. The angle bisector $AI$ meets $BC$ at $D$ and circumcircle of $\Delta ABC$ at $E$. The line $EI'$ meets the circumcircle at $F$. Prove, that,
a) $ \frac{AI}{IE}=\frac{ID}{DE}$
b) $ IA=IF$

Let $ABC$ be a right triangle, with the right angle at $A$. The altitude from $A$ meets $BC$ at $H$ and $M$ is the midpoint of the hypotenuse $[BC]$. On the legs, in the exterior of the triangle, equilateral triangles $BAP$ and $ACQ$ are constructed. If $N$ is the intersection point of the lines $AM$ and $PQ$, prove that the angles $\angle NHP$ and $\angle AHQ$ are equal.
Miguel Ochoa Sanchez and Leonard Giugiuc
2017 Romania JBMO TST 5.2 (Cuba 2003)
Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle  PEQ = 2\angle APB$

2018 Romania JBMO TST 1.3
Let $ABC$ be a triangle with $AB > AC$. Point $P \in (AB)$ is such that $\angle ACP = \angle ABC$. Let $D$ be the reflection of $P$ into the line $AC$ and let $E$ be the point in which the circumcircle of $BCD$ meets again the line $AC$. Prove that $AE = AC$.

 2018 Romania JBMO TST 2.3
Triangle $ABC$ has the property that there exists a unique point $X$ on the line segment $BC$ such that $AX^2 = BX \cdot  CX$. Prove that $AB + AC = BC\sqrt2$

 2018 Romania JBMO TST 3.2 (Caucasus 2017 9.4)
In an acute traingle $ABC$ with  $AB< BC$ let $BH_b$ be its altitude, and let $O$ be the circumcenter. A line through $H_b$ parallel to $CO$ meets $BO$ at $X$. Prove that $X$ and the midpoints of $AB$ and $AC$ are collinear.

Let $ABC$ be an acute triangle, with $AB \ne AC$. Let $D$ be the midpoint of the line segment $BC$, and let $E$ and $F$ be the projections of $D$ onto the sides $AB$ and $AC$, respectively. If $M$ is the midpoint of the line segment $EF$, and $O$ is the circumcenter of triangle $ABC$, prove that the lines $DM$ and $AO$ are parallel.

2018 Romania JBMO TST 4.3
Let $ABCD$ be a cyclic quadrilateral. The line parallel to $BD$ passing through $A$ meets the line parallel to $AC$ passing through $B$ at $E$. The circumcircle of triangle $ABE$ meets the lines $EC$ and $ED$, again, at $F$ and $G$, respectively. Prove that the lines $AB, CD$ and $FG$ are either parallel or concurrent.
2018 Romania JBMO TST 5.4
Let $ABC$ be a triangle, and let $E$ and $F$ be two arbitrary points on the sides $AB$ and $AC$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ again at point $M$. Let $D$ be the reflection of point $M$ across the line $EF$ and let $O$ be the circumcenter of triangle $ABC$. Prove that $D$ is on $BC$ if and only if $O$ belongs to the circumcircle of triangle $AEF$.
2018 Romania JBMO TST 6.3 (All Russian 2015 9.7) (also)
An acute-angled $ABC \ (AB<AC)$ is inscribed into a circle $\omega$. Let $M$ be the centroid of $ABC$, and let $AH$ be an altitude of this triangle. A ray $MH$ meets $\omega$ at $A'$. Prove that the circumcircle of the triangle $A'HB$ is tangent to $AB$.  

2019 Romania JBMO TST 1.3
Let $ABC$ a triangle, $I$ the incenter, $D$ the contact point of the incircle with the side $BC$ and $E$ the foot of the bisector of the angle $A$. If $M$ is the midpoint of the arc $BC$ which contains the point $A$ of the circumcircle of the triangle $ABC$ and $\{F\} = DI \cap AM$, prove that $MI$ passes through the midpoint of $[EF]$.

2019 Romania JBMO TST 2.3
Let $d$ be the tangent at $B$ to the circumcircle of the acute scalene triangle $ABC$. Let $K$ be the orthogonal projection of the orthocenter, $H$, of triangle $ABC$ to the line $d$ and $L$ the midpoint of the side $AC$. Prove that the triangle $BKL$ is isosceles.

2019 Romania JBMO TST 3.3
A circle with center $O$ is internally tangent to two circles inside it at points $S$ and $T$. Suppose the two circles inside intersect at $M$ and $N$ with $N$ closer to $ST$. Show that $OM$ and $MN$ are perpendicular if and only if $S,N, T$ are collinear.

2019 Romania JBMO TST 4.2 (IMO SL 2004)
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.

by Hojoo Lee, Korea
2019 Romania JBMO TST 5.3
In the acute triangle $ABC$ point $I$ is the incenter, $O$ is the circumcenter, while $I_a$ is the excenter opposite the vertex $A$. Point $A'$ is the reflection of $A$ across the line $BC$. Prove that angles $\angle IOI_a$ and \angle $IA'I_a$ are equal.

2019 Romania JBMO TST 6.3
Let $ABC$ be a triangle in which $AB < AC, D$ is the foot of the altitude from $A, H$ is the orthocenter, $O$ is the circumcenter, $M$ is the midpoint of the side $BC, A'$ is the reflection of $A$ across $O$, and $S$ is the intersection of the tangents at $B$ and $C$ to the circumcircle. The tangent at $A'$ to the circumcircle intersects $SC$ and $SB$ at $X$ and $Y$ , respectively. If $M,S,X,Y$ are concyclic, prove that lines $OD$ and $SA'$ are parallel.

The incircle of triangle $ABC$ is tangent to the sides $AB,AC$ and $BC$ at the points $M,N$ and $K$ respectively. The median $AD$ of the triangle $ABC$ intersects $MN$ at the point $L$. Prove that $K,I$ and $L$ are collinear, where $I$ is the incenter of the triangle $ABC$.

Let $I$ be the incenter of triangle $ABC$. The circle of centre $A$ and radius $AI$ intersects the circumcircle of triangle $ABC$ in $M$ and $N$. Prove that the line $MN$ is tangent to the incircle of triangle $ABC$

Let $ABC$ be a triangle such that $\angle A=30^\circ$ and $\angle B=80^\circ$. Let $D$ and $E$ be points on sides $AC$ and $BC$ respectively so that $\angle ABD=\angle DBC$ and $DE\parallel AB$. Determine the measure of $\angle EAC$.

Let $ABC$ be a right triangle $(AB<AC)$ with heights $AD, BE,$ and $CF$ and orthocenter $H$. Let $M$ denote the midpoint of $BC$ and let $X$ be the second intersection of the circle with diameter $HM$ and line $AM.$ Given that lines $HX$ and $BC$ intersect at $T,$ prove that the circumcircles of $\triangle TFD$ and $\triangle AEF$ are tangent.

Let $M,N$ and $P$ be the midpoints of sides $BC,CA$ and $AB$ respectively, of the acute triangle $ABC.$ Let $A',B'$ and $C'$ be the antipodes of $A,B$ and $C$ in the circumcircle of triangle $ABC.$ On the open segments $MA',NB'$ and $PC'$ we consider points $X,Y$ and $Z$ respectively such that\[\frac{MX}{XA'}=\frac{NY}{YB'}=\frac{PZ}{ZC'}.\]
Prove that the lines $AX,BY,$ and $CZ$ are concurrent at some point $S.$
Prove that $OS<OG$ where $O$ is the circumcenter and $G$ is the centroid of triangle $ABC.$

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two circles, internally tangent at $P$ ($\mathcal{C}_2$ lies inside of $\mathcal{C}_1$). A chord $AB$ of $\mathcal{C}_1$ is tangent to $\mathcal{C}_2$ at $C.$ Let $D$ be the second point of intersection between the line $CP$ and $\mathcal{C}_1.$ A tangent from $D$ to $\mathcal{C}_2$ intersects $\mathcal{C}_1$ for the second time at $E$ and it intersects $\mathcal{C}_2$ at $F.$ Prove that $F$ is the incenter of triangle $ABE.$

Let $ABC$ be an acute scalene triangle. Let $D$ be the foot of the $A$-bisectrix and $E$ be the foot of the $A$-altitude. The perpendicular bisector of the segment $AD$ intersects the semicircles of diameter $AB$ and $AC$ which lie on the outside of triangle $ABC$ at $X$ and $Y$ respectively. Prove that the points $X,Y,D$ and $E$ lie on a circle.



ROMOP 2011 Juniors - Romanian MOP


Let $P$ be a point inside the triangle $A B C$, and the $A_1, B_1, C_1$ intersections of the lines $AP, BP, CP$ with the sides $BC, CA, AB$ respectively. Prove that the angle $\angle C_1A_1B_1$ is right if and only if $(A_1B_1$ is the bisector of the angle $\angle AA_1C$, and $(A_1C_1$is the bisector of the angle $\angle AA_1B$.

On the sides $(BC), (CA), (AB)$ of the triangle $A B C$ are considered respectively the points $D, E, F$. Points $P, Q, R$ are points of intersection with the circle circumscribed triangle $A B C$ of straight lines $AD, BE, CF$, different than $A,B,C$. Prove the inequality $\frac {AD} {PD} + \frac {BE} {QE} + \frac {CF} {RF} \ge 9$.

Give a triangle$ A B C$ in which the $BC> \max \{AB, AC \}$ are points $P \in (AB), Q \in (AC)$ so that $\angle PCB = \angle BAC$ and $\angle QBC = \angle BAC$. Prove that the line determined by the centers of the circles circumscribed by the triangles $A B C$ and $APQ$ is perpendicular to $BC$.

A point $P$ of the circle circumscribed to the triangle $A B C$ is projected on the lines of the sides $[BC] $ and $[AC]$ as the points $D$ and $E$ respectively. We also denote $L,M$ the midpoints of the segments $[D], [BE]$ respectively. Show that $LM \perp DE$.

On the sides of the triangle $A B C$, the isoscelestrianglesare built on the outside, $ABF, BCD, CAE$ based on the sides of the triangle $A B C$. Prove that the perpendiculars from $A,B,C$ respectively on $EF, FD, DE$ are concurrent.

In the acute triangle $ABC$ with $AB \neq AC$ , let $BB'$ and $CC'$ be the altitudes . Let $M$ be the midpoint middle $[BC]$, $ H$ the orthocenter of the triangle $ABC$ and $\{D\}=B'C'  \cap BC$. Prove that $DH \perp AM$.

Let $M$ be the midpoint of the side $[BC]$ of the triangle $A B C$, $I$ it's incenter and $H$ it's orthocenter. The line $MI$ intersects $AH$ in $E$. Prove that $AE = r$, where $r$ is the inradius of $ABC$.

Let $P$ be a point located inside the side $a$ of equilateral triangle $A B C$ and let $O$ be it's circumcenter. If $OP = d$, then the area of the triangle of sides of equal lengths $MA, MB, MC$ is given by $S = \frac {\sqrt 3} {12} | a ^ 2-3d ^ 2 |$.

Consider the triangle $A B C$ in which $(AD$ is the bisector of the angle $BAC, D \in (BC)$. The circle that contains the point $A$ and is tangent to the line $BC$ at the point $D$ intersects the segments $[AB]$ and $[AC]$ at the points $E,$ respectively and the segment $(BF)$ at the point $N$. Prove that the midpoint of the segment $[BD]$ is located on the line $AN$.

Conside a trapezoid $A B C D$ inscribed in the circle $C(O,R)$. Let $ \{M \} = AC \cap BD$ and $\{T \} = AB \cap DC$. The parallel through the point $M$ at the line $BC$ intersects the circle $C(O,R)$ at the points $E$ and $F$. Prove that the line $ TE$ is tangent to the circle $C(O,R)$.

On the circle $C(O,R)$, points $ A,B,C,D$ and $E$ are considered in this order, so that the $AC \parallel DE$, point $B$ belongs to the arc $\stackrel {\frown} {AC}$, $M$ is the midpoint of the segment $[BD]$ and also $\angle AMB=\angle BMC$. Prove that the midpoint of the segment $[AC]$ is on the line $BE$.

Consider the triangle $ABC$ with $AB<AC$ and the point $D\in(BC)$ such that $AD\perp BC$. Prove that exists only one point $M\in AD, M \ne A$ so that $\angle MBA= \angle MCA$.

In the triangle the $A B C$, consider altitude $AD, D \in (BC)$. The points $I_ {1},I_ {2}$ are the centers of the circles inscribed in the triangles $ABD, ACD$ respectively . The line $I_  {1}I_ {2}$ intersects the segments $(AB)$ and $(AC)$ at the points $E$ and $F$. Prove that the triangle $AEF$ is isosceles if and only if the triangle $A B C$ is isosceles or right.

Let $ABC$ be a triangle for which denote the midpoints $D\in (BC)$ , $E\in (CA)$ , $F\in (AB)$ of the mentioned sides.
Suppose that exist two points $M\in (AB)$ , $N\in (AC)$ so that $AM^{2}+AN^{2}=BM^{2}+CN^{2}$ , $M\ne F$ or
$N\ne E$ and $m\left(\widehat{MDN}\right)=A$ . Ascertain the value $A$ of the angle $\angle BAC$

Let $E,F$ be the midpoints of the sides $[BC], [CD]$ respectively of the convex quadrilateral $ABCD$. The segments [AE], $[AF] $and $[EF]$ cut $ABCD$ into four triangles whose areas are four consecutive non-zero natural numbers. Find the maximum area of the triangle $BAD$.

sources:
pregatirematematicaolimpiadejuniori.wordpress.com/ 

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