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Romania District 2001-19 VII+ 97p

geometry problems from Romanian Nationan Mathematical Olympiads - District Round
with aops links in the names

 problems collected inside aops

2001 - 2019, grades VII - XII 


2000 Romania District IX P3
Let $ABCD$ be a parallelogram and  $M$ be a point on it's plane. Prove that $\overrightarrow {MA}+\overrightarrow{MC} =\overrightarrow{MB}+\overrightarrow {MD}$

Let be a circle centeted at $ O, $ and $ A,B,C, $ points situated on this circle. Show that if
$ \left|\overrightarrow{OA} +\overrightarrow{OB}\right| = \left|\overrightarrow{OB} +\overrightarrow{OC}\right| = \left|\overrightarrow{OC} +\overrightarrow{OA}\right| , $ then $ A=B=C, $ or $ ABC $ is an equilateral triangle.
Let $ \alpha $ be a plane and let $ ABC $ be an equilateral triangle situated on a parallel plane whose distance from $ \alpha $ is $ h. $ Find the locus of the points $ M\in\alpha $ for which $ \left|MA\right| ^2 +h^2 = \left|MB\right|^2 +\left|MC\right|^2. $

Consider the pyramid $ VABCD, $ where $ V $ is the top and $ ABCD $ is a rectangular base. If $ \angle BVD = \angle AVC, $ then prove that the triangles $ VAC $ and $ VBD $ share the same perimeter and area.
Consider a triangle $\Delta ABC$ and three points $D,E,F$ such that: $B$ and $E$ are on different side of the line $AC$, $C$ and $D$ are on different sides of $AB$, $A$ and $F$ are on the same side of the line $BC$. Also $\Delta ADB \sim \Delta CEA \sim \Delta CFB$. Let $M$ be the middle point of $AF$. Prove that:
a) $\Delta BDF \sim \Delta FEC$.
b) $M$ is the middle point of $DE$.
Dan Branzei 
Consider a convex qudrilateral $ABCD$ and $M\in (AB),\ N\in (CD)$ such that $\frac{AM}{BM}=\frac{DN}{CN}=k$. Prove that $BC\parallel AD$ if and only if
$MN=\frac{1}{k+1} AD+\frac{k}{k+1} BC$
Consider four points $A,B,C,D$ not in the same plane such that
$AB=BD=CD=AC=\sqrt{2} AD=\frac{\sqrt{2}}{2}BC=a$
Prove that:
a) There is a point $M\in [BC]$ such that $MA=MB=MC=MD$.
b) $2m(\sphericalangle(AD,BC))=3m(\sphericalangle((ABC),(BCD)))$
c) $6(d(A,CD))^2=7(d(A,(BCD)))^2$
Consider a rectangular parallelepiped $ABCDA'B'C'D'$ in which we denote $AB=a,\ BC=b,\ AA'=c$. Let $DE\perp AC,\ DF\perp A'C,\ E\in AC,\ F \in A'C$ and $C'P\perp B'D',\ C'Q\perp BD',\ P\in B'D',\ Q\in BD'$. Prove that the planes $(DEF)$ and $(C'PQ)$ are perpendicular if and only if $a^2+c^2=b^2$.

2001 Romania District X P3
Consider an inscriptible polygon $ABCDE$. Let $H_1,H_2,H_3,H_4,H_5$ be the orthocenters of the triangles $ABC,BCD,CDE,DEA,EAB$ and let $M_1,M_2,M_3,M_4,M_5$ be the midpoints of $DE,EA,AB,BC$ and $CD$, respectively. Prove that the lines $H_1M_1,H_2M_2,H_3M_3,H_4M_4,H_5M_5$ have a common point.

Consider the equilateral triangle $ABC$ with center of gravity $G$. Let $M$ be a point, inside the triangle and $O$ be the midpoint of the segment $MG$. Three segments go through $M$, each parallel to one side of the triangle and with the ends on the other two sides of the given triangle.
a) Show that $O$ is at equal distances from the midpoints of the three segments considered.
b) Show that the midpoints of the three segments are the vertices of an equilateral triangle.
Given the rectangle $ABCD$. The points $E ,F$  lie on the segments $(BC) , (DC)$ respectively, such that $\angle DAF = \angle FAE$. Proce  that if $DF + BE = AE$ then $ABCD$ is square.

2002 Romania District VIII P3
Consider the regular pyramid $VABCD$  with the vertex in $V$ which measures the angle formed by two opposite lateral edges is $45^o$. The points $M,N,P$ are respectively, the projections of the point $A$ on the line $VC$, the symmetric of the point $M$ with respect to the plane $(VBD)$ and the symmetric of the point $N$ with respect to $O$. ($O$ is the center of the base of the pyramid.)
a) Show that the polyhedron $MDNBP$ is a regular pyramid.
b) Determine the measure of the angle between the line $ND$ and the plane $(ABC) $ 

The cube $ABCDA' B' C' D' $has of length a. Consider the points $K \in [AB], L  \in [CC' ],  M \in [D'A']$.
a) Show that $\sqrt3 KL>\ge  KB + BC + CL$
b) Show that the perimeter of triangle $KLM$ is strictly greater than  $2a\sqrt3$.

Let $ ABCD $ be an inscriptible quadrilateral and $ M $ be a point on its circumcircle, distinct from its vertices. Let $ H_1,H_2,H_3,H_4 $ be the orthocenters of $ MAB,MBC, MCD, $ respectively, $ MDA, $ and $ E,F, $ the midpoints of the segments $ AB, $ respectivley, $ CD. $ Prove that:
a) $ H_1H_2H_3H_4 $ is a parallelogram.
b) $ H_1H_3=2\cdot EF. $

2002 Romania District IX P3
Let $ G $ be the center of mass of a triangle $ ABC, $ and the points $ M,N,P $ on the segments $ AB,BC, $ respectively, $ CA $ (excluding the extremities) such that
$ \frac{AM}{MB} =\frac{BN}{NC} =\frac{CP}{PA} . $ $ G_1,G_2,G_3 $ are the centers of mass of the triangles $ AMP, BMN, $ respectively, $ CNP. $ Pove that:
a) The centers of mas of $ ABC $ and $ G_1G_2G_3 $ are the same.
b) For any planar point $ D, $ the inequality $ 3\cdot DG< DG_1+DG_2+DG_3<DA+DB+DC $ holds.

In the right triangle $ABC$ ( $\angle A = 90^o$),  $D$ is the intersection of the bisector of the angle $A$ with the side $(BC)$, and $P$ and $Q$ are the projections of the point $D$ on the sides $(AB),(AC)$  respectively . If $BQ \cap DP=\{M\}$, $CP \cap DQ=\{N\}$, $BQ\cap CP=\{H\}$, show that:
a) $PM = DN$
b) $MN \parallel BC$
c) $AH \perp BC$.

Let $ABC$ be a triangle. Let $B'$ be the symmetric of $B$ with respect to $C, C'$ the symmetry of $C$ with respect to $A$ and $A'$ the symmetry of $A$ with respect to $B$.
a) Prove that the area of ​​triangle $AC'A'$  is twice the area of ​​triangle $ABC$.

b) If we delete points $A, B, C$, how can they be reconstituted? Justify your reasoning.

Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12  AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$. 

a) Let $MNP$ be a triangle such that $\angle MNP> 60^o$. Show that the side $MP$ cannot be the smallest side of the  triangle $MNP$.
b) In a plane the equilateral triangle $ABC$ is considered. The point $V$ that does not belong to the plane $(ABC)$ is chosen so that $\angle VAB = \angle VBC = \angle VCA$. Show that if $VA = AB$, the tetrahedron $VABC$ is regular.
Valentin Vornicu
On a board are drawn the points $A,B,C,D$. Yetti constructs the points $A^\prime,B^\prime,C^\prime,D^\prime$ in the following way: $A^\prime$ is the symmetric of $A$ with respect to $B$, $B^\prime$ is the symmetric of $B$ wrt $C$, $C^\prime$ is the symmetric of $C$ wrt $D$ and $D^\prime$ is the symmetric of $D$ wrt $A$.Suppose that Armpist erases the points $A,B,C,D$. Can Yetti rebuild them?

(a) If $\displaystyle ABC$ is a triangle and $\displaystyle M$ is a point from its plane, then prove that $ \displaystyle AM \sin A \leq BM \sin B + CM \sin C . $
(b) Let $\displaystyle A_1,B_1,C_1$ be points on the sides $\displaystyle (BC),(CA),(AB)$ of the triangle $\displaystyle ABC$, such that the angles of $\triangle A_1 B_1 C_1$ are $\widehat{A_1} = \alpha, \widehat{B_1} = \beta, \widehat{C_1} = \gamma$. Prove that $ \displaystyle \sum A A_1 \sin \alpha \leq \sum BC \sin \alpha . $
Let $ABC$ be a triangle and $D$ a point on the side $BC$. The angle bisectors of $\angle ADB ,\angle ADC$ intersect $AB ,AC$ at points $M ,N$ respectively. The angle bisectors of $\angle ABD , \angle ACD$ intersects $DM , DN$ at points $K , L$  respectively. Prove that $AM = AN$ if and only if $MN$ and $KL$ are parallel.

2004 Romania District VII P4
Consider the isosceles right triangle $ABC$ ($AB = AC$) and the points $M, P \in  [AB]$ so that $AM = BP$. Let $D$ be the midpoint of the side $BC$ and $R, Q$ the intersections of the perpendicular from A on$ CM$ with $CM$ and $BC$ respectively. Prove that
a) $\angle  AQC =  \angle PQB$
b)  $\angle DRQ  = 45^o$

In the right trapezoid $ABCD$ with $AB \parallel CD,  \angle B = 90^o$ and $AB = 2DC$.
At points $A$ and $D$ there is therefore a part of the plane $(ABC)$ perpendicular to the plane of the trapezoid, on which the points $N$ and $P$ are taken, ($AP$ and $PD$ are perpendicular to the plane) such that  $DN = a$ and $AP = \frac{a}{2}$ . Knowing that M is the midpoint of the side BC and the triangle $MNP$ is equilateral, determine:
a) the cosine of the angle between the planes $MNP$ and $ABC$.
b) the distance $a$ from $D$ to the plane $MNP$

On the tetrahedron $ ABCD $ make the notation $ M,N,P,Q, $ for the midpoints of $ AB,CD,AC, $ respectively, $ BD. $ Additionally, we know that $ MN $ is the common perpendicular of $ AB,CD, $ and $ PQ $ is the common perpendicular of $ AC,BD. $ Show that $ AB=CD, BC=DA, AC=BD. $

Let $ABC$ be a triangle and let $M$ be the midpoint of the side $AB$. Let $BD$ be the interior angle bisector of $\angle ABC$, $D\in AC$. Prove that if $MD \perp BD$ then  $AB=3BC$.
In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.
Let $ABCD$ and $ABEF$ be two squares situated in two perpendicular planes and let $O$ be the intersection of the lines $AE$ and $BF$. If $AB=4$ compute:
a) the distance from $B$ to the line of intersection between the planes $(DOC)$ and $(DAF)$;
b) the distance between the lines $AC$ and $BF$.
Prove that if the circumcircles of the faces of a tetrahedron $ABCD$ have equal radii, then $AB=CD$, $AC=BD$ and $AD=BC$.

Let $ABC$ be a triangle inscribed in a circle of center $O$ and radius $R$. Let $I$ be the incenter of $ABC$, and let $r$ be the inradius of the same triangle, $O\neq I$, and let $G$ be its centroid. Prove that $IG\perp BC$ if and only if $b=c$ or $b+c=3a$.


Let $ABC$ be a non-right-angled triangle and let $H$ be its orthocenter. Let $M_1,M_2,M_3$ be the midpoints of the sides $BC$, $CA$, $AB$ respectively. Let $A_1$, $B_1$, $C_1$ be the symmetrical points of $H$ with respect to $M_1$, $M_2$ and $M_3$ respectively, and let $A_2$, $B_2$, $C_2$ be the orthocenters of the triangles $BA_1C$, $CB_1A$ and $AC_1B$ respectively. Prove that:
a) triangles $ABC$ and $A_2B_2C_2$ have the same centroid;
b) the centroids of the triangles $AA_1A_2$, $BB_1B_2$, $CC_1C_2$ form a triangle similar with $ABC$.

Let $O$ be a point equally distanced from the vertices of the tetrahedron $ABCD$. If the distances from $O$ to the planes $(BCD)$, $(ACD)$, $(ABD)$ and $(ABC)$ are equal, prove that the sum of the distances from a point $M \in \textrm{int}[ABCD]$, to the four planes, is constant.

 2006 Romania District VII P2
In triangle $ABC$ we have $\angle ABC = 2 \angle ACB$. Prove that
a) $AC^2 = AB^2 + AB \cdot BC$;
b) $AB+BC < 2 \cdot AC$.

2006 Romania District VII P4
Let $ABC$ be a triangle with $AB=AC$. Let $D$ be the midpoint of $BC$, $M$ the midpoint of $AD$ and $N$ the foot of the perpendicular from $D$ to $BM$. Prove that $\angle ANC = 90^\circ$.

2006 Romania District VIII P1
On the plane of triangle $ABC$ with $\angle BAC = 90^\circ$ we raise perpendicular lines in $A$ and $B$, on the same side of the plane. On these two perpendicular lines we consider the points $M$ and $N$ respectively such that $BN < AM$. Knowing that $AC = 2a$, $AB = a\sqrt 3$, $AM=a$ and that the plane $MNC$ makes an angle of $30^\circ$ with the plane $ABC$ find
a) the area of the triangle $MNC$,
b) the distance from $B$ to the plane $MNC$.

Let $ABCD$ be a convex quadrilateral, $M$ the midpoint of $AB$, $N$ the midpoint of $BC$, $E$ the intersection of the segments $AN$ and $BD$, $F$ the intersection of the segments $DM$ and $AC$. Prove that if $BE = \frac 13 BD$ and $AF = \frac 13 AC$, then $ABCD$ is a parallelogram.

2006 Romania District X P2
Let $ABC$ be a triangle and let $M,N,P$ be points on the sides $BC$, $CA$ and $AB$ respectively such that $ \frac{AP}{PB} = \frac{BM}{MC} = \frac{CN}{AN}. $ Prove that triangle if $MNP$ is equilateral then triangle $ABC$ is equilateral.

 2007 Romania District VII P1
Point $O$ is the intersection of the perpendicular bisectors of the sides of the triangle $\vartriangle ABC$ . Let $D$ be the intersection of the line $AO$ with the segment $[BC]$. Knowing that $OD = BD = \frac 13  BC$, find the measures of the angles of the triangle $\vartriangle ABC$.
2007 Romania District VIII P2
Consider a rectangle $ABCD$ with $AB = 2$  and  $BC = \sqrt3$. The point $M$ lies on the side $AD$ so that $MD = 2  AM$ and the point $N$ is the midpoint of the segment $AB$. On the plane of the rectangle rises the perpendicular MP and we choose the point $Q$ on the segment $MP$ such that  the measure of the angle between the planes $(MPC)$ and $(NPC)$ shall be $45^o$, and the measure of the angle between the planes $(MPC)$ and $(QNC)$ shall be $60^o$.
a) Show that the lines $DN$ and $CM$ are perpendicular.
b) Show that the point $Q$ is the midpoint of the segment $MP$.

Consider $ \triangle ABC$ and points $ M \in (AB)$, $ N \in (BC)$, $ P \in (CA)$, $ R \in (MN)$, $ S \in (NP)$, $ T \in (PM)$ such that $ \frac {AM}{MB} = \frac {BN}{NC} = \frac {CP}{PA} = k$ and $ \frac {MR}{RN} = \frac {NS}{SP} = \frac {PT}{TN} = 1 - k$ for some $ k \in (0, 1)$. Prove that $ \triangle STR \sim \triangle ABC$ and, furthermore, determine $ k$ for which the minimum of $ [STR]$ is attained

Let $ABC$ be a triangle with $BC=a$ $AC=b$ $AB=c$. For each line $\Delta$ we denote $d_{A}, d_{B}, d_{C}$ the distances from $A,B, C$ to $\Delta$ and we consider the expresion $E(\Delta)=ad_{A}^{2}+bd_{B}^{2}+cd_{C}^{2}$. Prove that if $E(\Delta)$ is minimum, then $\Delta$ passes through the incenter of $\Delta ABC$.

 2008 Romania District VII P2
Consider the square $ABCD$ and $E \in (AB)$. The diagonal $AC$ intersects the segment $[DE]$ at point $P$. The perpendicular taken from point $P$ on $DE$ intersects the side $BC$ at point $F$. Prove that $EF = AE + FC$.
2008 Romania District VIII P1
A regular tetrahedron is sectioned with a plane after a rhombus. Prove that the rhombus is square.
2008 Romania District VIII P3  (Gazeta Matematica, 2007)
Let $ABCDA' B' C' D '$ be a cube  , $M$ the foot of the perpendicular from $A$ on the plane $(A'CD)$, $N$ the foot of the perpendicular from $B$ on the diagonal $A'C$ and $P$ is symmetric of the point $D$ with respect to $C$. Show that the points $M, N, P$ are collinear.

Let $ ABCD$ be a cyclic quadrilater. Denote $ P=AD\cap BC$ and $ Q=AB \cap CD$. Let $ E$ be the fourth vertex of the parallelogram $ ABCE$ and $ F=CE\cap PQ$. Prove that $ D,E,F$ and $ Q$ lie on the same circle.

 2009 Romania District VII P4
Let $ABC$ be an equilateral $ABC$. Points $M, N, P$ are located on the sides $AC, AB, BC$, respectively,  such that $\angle CBM= \frac{1}{2} \angle AMN = \frac{1}{3} \angle BNP$ and  $\angle CMP = 90 ^o$.
a) Show that $\vartriangle NMB$ is isosceles.
b) Determine $\angle CBM$.

2009 Romania District VIII P3
Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and  $(HBS)$.

On the sides $ AB $ and $ AC $ of the triangle $ ABC $ consider the points $ D, $ respectively, $ E, $ such that $ \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} . $ If $ T $ is the intersection of $ DC $ and $ BE, $ determine the real number $ \alpha $ so that: $ \overrightarrow{TB} +\overrightarrow{TC} =\alpha\cdot\overrightarrow{TA} . $

a) Let $ z_1,z_2,z_3 $ be three complex numbers of same absolute value, and $ 0=z_1+z_2+z_3. $ Show that these represent the affixes of an equilateral triangle.
b) Find all subsets formed by roots of the same unity that have the property that any three elements of every such, doesn’t represent the vertices of an equilateral triangle.

 2010 Romania District VII P3
Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.

2010 Romania District VII P4
We consider the quadrilateral $ABCD$, with $AD = CD = CB$ and $AB \parallel CD$. Points $E$ and $F$ belong to the segments $CD$ and $CB$ so that angles $\angle ADE = \angle AEF$. Prove that:
a) $4CF \le CB$ ,
b) if $4CF = CB$, then $AE$ is the bisector of the angle $\angle DAF$.

2010 Romania District VIII P3
Consider the cube $ABCDA'B'C'D'$. The bisectors of the angles $\angle A' C'A$ and $\angle A' AC'$ intersect $AA'$ and $A'C$ in the points $P$, respectively $S$. The point $M$ is the foot of the perpendicular from $A'$ on $CP$ , and $N$ is the foot of the perpendicular from $A'$  to $AS$. Point $O$ is the center of the face $ABB'A'$
a) Prove that the planes $(MNO)$ and $(AC'B)$ are parallel.
b) Calculate the distance between these planes, knowing that $AB = 1$.

A right that passes through the incircle $ I$ of the triangle $ \Delta ABC$ intersects the side $ AB$ and $ CA$ in $ P$, respective $ Q$. We denote $ BC=a\ , \ AC=b\ ,\ AB=c$ and $ \frac{PB}{PA}=p\ ,\ \frac{QC}{QA}=q$.
i) Prove that: $ a(1+p)\cdot \overrightarrow{IP}=(a-pb)\overrightarrow{IB}-pc\overrightarrow{IC}$
ii) Show that $ a=bp+cq$.
iii) If $ a^2=4bcpq$, then the rights $ AI\ ,\ BQ$ and $ CP$ are concurrent

The isosceles trapezoid $ABCD$ has perpendicular diagonals. The parallel to the bases through the point of intersection of the diagonals intersects the non-parallel sides $[BC]$ and $[AD]$ in the points $P$, respectively $R$. The point $Q$ is symmetric of the point $P$ with respect to the middle of the segment $[BC]$. Prove that:
a) $QR = AD$,
b) $QR \perp AD$.

Let $ABCA'B'C'$  a right triangular prism with the bases equilateral triangles. A plane $\alpha$ containing point $A$ intersects the rays $BB'$ and  $CC'$  at points E and $F$, so that $S_ {ABE} + S_{ACF} = S_{AEF}$. Determine the measure of the angle formed by the plane $(AEF)$ with the plane $(BCC')$.

On the sides $ AB,BC,CD,DA $ of the parallelogram $ ABCD, $ consider the points $ M,N,P, $ respectively, $ Q, $ such that $ \overrightarrow{MN} +\overrightarrow{QP} =\overrightarrow{AC} . $ Show that $ \overrightarrow{PN} +\overrightarrow{QM} = \overrightarrow{DB} . $

Let $ABC$ be a sharp triangle. Consider the points $M, N \in (BC), Q \in (AB), P \in (AC)$ such that the $MNPQ$ is a rectangle. Prove that if the center of the rectangle $MNPQ$ coincides with the center of gravity of the triangle $ABC$, then $AB = AC = 3AP$  
2012 Romania District VII P4
Consider the square $ABCD$ and the point $E$ on the side $AB$. The line $DE$ intersects the line $BC$ at point $F$, and the line $CE$ intersects the line $AF$ at point $G$. Prove that the lines $BG$ and $DF$ are perpendicular.

The pyramid  $VABCD$ has base the rectangle ABCD, and the side edges are congruent. Prove that the plane $(VCD)$ forms congruent angles with the planes $(VAC)$ and $(BAC)$ if and only if $\angle VAC = \angle BAC $.

A circle that passes through the vertices $ B,C $ of a triangle $ ABC, $ cuts the segments $ AB,AC $ (endpoints excluded) in $ N, $ respectively, $ M. $ Consider the point  $ P $ on the segment $ MN $ and $ Q $ on the segment $ BC $ (endpoints excluded on both segments) such that the angles $ \angle BAC,\angle PAQ $ have the same bisector. Show that:
a) $ \frac{PM}{PN} =\frac{QB}{QC} . $
b) The midpoints of the segments $ BM,CN,PQ $ are collinear.

2013 Romania District VII P3
On the sides $(AB)$ and $(AC)$ of the triangle $ABC$ are considered the points $M$ and $N$ respectively so that $ \angle ABC =\angle ANM$. Point $D$ is symmetric of point $A$ with respect to $B$, and $P$ and $Q$ are the midpoints of the segments $[MN]$ and $[CD]$, respectively.  Prove that the points $A, P$ and $Q$ are collinear if and only if $AC = AB \sqrt {2}$

2013 Romania District VII P4
Consider the square $ABCD$ and the point $E$ inside the angle $CAB$, such that $\angle BAE =15^o$, and the lines $BE$ and $BD$ are perpendicular. Prove that $AE = BD$.

Let be the regular hexagonal prism $ABCDEFA'B C'D'E'F'$ with the base edge of $12$ and the height of $12 \sqrt{3}$. We denote by $N$ the middle of the edge $CC'$.
a) Prove that the lines $BF'$ and $ND$ are perpendicular
b) Calculate the distance between the lines $BF'$ and $ND$.
2013 Romania District VIII P4
Consider a tetrahedron $ABCD$ in which $AD \perp BC$ and $AC \perp BD$. We denote by $E$ and $F$ the projections of point $B$ on the lines $AD$ and $AC$, respectively. If $M$ and $N$ are the midpoints of the segments $[AB]$ and $[CD]$, respectively, show that $MN \perp EF$

Given triangle $ABC$ and the points$D,E\in \left( BC \right)$, $F,G\in \left( CA \right)$, $H,I\in \left( AB \right)$ so that $BD=CE$, $CF=AG$ and $AH=BI$. Note with $M,N,P$ the midpoints of $\left[ GH \right]$, $\left[ DI \right]$ and $\left[ EF \right]$ and with ${M}'$ the intersection of the segments $AM$and $BC$.
a) Prove that $\frac{B{M}'}{C{M}'}=\frac{AG}{AH}\cdot \frac{AB}{AC}$.
b) Prove that the segments$AM$, $BN$ and $CP$ are concurrent.

Let $ABC$ be a triangle in which $\measuredangle{A}=135^{\circ}$. The perpendicular to the line $AB$ erected at $A$ intersects the side $BC$ at $D$, and the angle bisector of $\angle B$ intersects the side $AC$ at $E$. Find the measure of $\measuredangle{BED}$.
Let $ABCD$ be a square and consider the points $K\in AB, L\in BC,$ and $M\in CD$ such that $\Delta KLM$ is a right isosceles triangle, with the right angle at $L$. Prove that the lines $AL$ and $DK$ are perpendicular to each other.
In the right parallelopiped $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$, with $AB=12\sqrt{3}$ cm and $AA^{\prime}=18$ cm, we consider the points $P\in AA^{\prime}$ and $N\in A^{\prime}B^{\prime}$ such that $A^{\prime}N=3B^{\prime}N$. Determine the length of the line segment $AP$ such that for any position of the point $M\in BC$, the triangle $MNP$ is right angled at $N$.
Let $ABCDEF$ be a regular hexagon with side length $a$. At point $A$, the perpendicular $AS$, with length $2a\sqrt{3}$, is erected on the hexagon's plane. The points $M, N, P, Q,$ and $R$ are the projections of point $A$ on the lines $SB, SC, SD, SE,$ and $SF$, respectively. 
Prove that the points $M, N, P, Q, R$ lie on the same plane.
Find the measure of the angle between the planes $(MNP)$ and $(ABC)$.

Let $ABC$ be a triangle and let the points $D\in BC, E\in AC, F\in AB$, such that $ \frac{DB}{DC}=\frac{EC}{EA}=\frac{FA}{FB} $
The half-lines $AD, BE,$ and $CF$ intersect the circumcircle of $ABC$ at points $M,N$ and $P$. Prove that the triangles $ABC$ and $MNP$ share the same centroid if and only if the areas of the triangles $BMC, CNA$ and $APB$ are equal.


The medians $AD, BE$ and $CF$ of triangle $ABC$ intersect at $G$. Let $P$ be a point lying in the interior of the triangle, not belonging to any of its medians. The line through $P$ parallel to $AD$ intersects the side $BC$ at $A_{1}$. Similarly one defines the points $B_{1}$ and $C_{1}$. Prove that $ \overline{A_{1}D}+\overline{B_{1}E}+\overline{C_{1}F}=\frac{3}{2}\overline{PG} $

On the segment $ AC $ of the triangle $ ABC, $ let $ M $ be the midpoint of it, and let $ N $ a point on $ AM, $ distinct from $ A $ and $ M. $ The parallel through $ N $ with respect to $ AB $ intersects $ BM $ on $ P, $ the parallel through $ M $ with respect to $ BC $ intersects $ BN $ on $ Q, $ and the parallel through $ N $ with respect to $ AQ $ intersects $ BC $ on $ S. $
Prove that $ PS $ and $ AC $ are parallel.
At the exterior of the square $ ABCD $ it is constructed the isosceles triangle $ ABE $ with $ \angle ABE=120^{\circ} . M $ is the intersection of the bisector line of the angle $ \angle EAB $ with its perpendicular that passes through $ B; N $ is the intersection of the $ AB $ with its perpendicular that passe through $ M; P $ is the intersection of $ CN $ with $ MB. $
If $ G $ is the center of gravity of the triangle $ ABE, $ prove that $ PG $ and $ AE $ are parallel..
Consider the rectangular parallelepiped $ ABCDA'B'C'D' $ and the point $ O $ to be the intersection of $ AB' $ and $ A'B. $ On the edge $ BC, $ pick a point $ N $ such that the plane formed by the triangle $ B'AN $ has to be parallel to the line $ AC', $ and perpendicular to $ DO'. $
Prove, then, that this parallelepiped is a cube.

Consider the parallelogram $ ABCD, $ whose diagonals intersect at $ O. $ The bisector of the angle $ \angle DAC $ and that of $ \angle DBC $ intersect each other at $ T. $ Moreover, $ \overrightarrow{TD} +\overrightarrow{TC} =\overrightarrow{TO} . $ Find the angles of the triangle $ ABT. $

 2016 Romania District VII P3
Let be a triangle $ ABC $ with $ \angle BAC = 90^{\circ } . $ On the perpendicular of $ BC $ through $ B, $ consider $ D $ such that $ AD=BC. $ Find $ \angle BAD. $

2016 Romania District VII P4
Consider the triangle $ ABC $ with $ \angle BAC>60^{\circ } $ and $ \angle BCA>30^{\circ } . $ On the other semiplane than that determined by $ BC $ and $ A $ we have the points $ D $ and $ E $ so that $ \angle ABE =\angle CBD =\angle BAE +30^{\circ } =\angle BCD +30^{\circ } =90^{\circ } . $ Note by $ F,H $ the midpoints of $ AE, $ respectively, $ CD, $ and with $ G $ the intersection of $ AC $ and $ DE. $ Show:
a) $ EBD\sim ABC $
b) $ FGH\equiv ABC $

2016 Romania District VIII P1
Let be a pyramid having a square as its base and the projection of the top vertex to the base is the center of the square. Prove that two opposite faces are perpendicular if and only if the angle between two adjacent faces is $ 120^{\circ } . $

2016 Romania District VIII P4
Let $ ABCDA’B’C’D’ $ a right parallelepiped and $ M,N $ the feet of the perpendiculars of $ BD $ through $ A’, $ respectively, $ C’. $ We know that $ AB=\sqrt 2, BC=\sqrt 3, AA’=\sqrt 2. $
a) Prove that $ A’M\perp C’N. $
b) Calculate the dihedral angle between the plane formed by $ A’MC $ and the plane formed by $ ANC’. $

Let $ ABCD $ be a sqare and $ E $ be a point situated on the segment $ BD, $ but not on the mid. Denote by $ H $ and $ K $ the orthocenters of $ ABE, $ respectively, $ ADE. $ Show that $ \overrightarrow{BH}=\overrightarrow{KD} . $

2016 Romania District X P2
Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying:
$ a^2+b^2+c^2-ab-bc-ca=0. $ Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.
On the side $ CD $ of the square $ ABCD, $ consider $ E $ for which $ \angle ABE =60^{\circ } . $ On the line $ AB, $ take the point $ F $ distinct from $ B $ such that $ BE=BF $ and such that it is on the segment $ AB, $ or $ A $ is on $ BF. $ Moreover, $ M $ is the intersection of $ EF,AD. $
a) Show that $ \angle BME =75^{\circ } . $
b) If the bisector of $ \angle CBE $ intersects $ CD $ in $ N, $ show that $ BMN $ is equilateral.

We have a triangle with $ \angle BAC=\angle BCA. $ The point $ E $ is on the interior bisector of $ \angle ABC $ so that $ \angle EAB =\angle ACB. $ Let $ D $ be a point on $ BC $ such that $ B $ is on the segment $ CD $ (endpoints excluded) and $ BD=AB. $ Show that the midpoint of $ AC $ is on the line $ DE. $
Let $ ABCDA’B’C’D’ $ a cube. $ M,P $ are the midpoints of $ AB, $ respectively, $ DD’. $
a) Show that $ MP, A’C $ are perpendicular, but not coplanar.
b) Calculate the distance between the lines above.
2017 Romania District VIII P4
An ant can move from a vertex of a cube to the opposite side of its diagonal only on the edges and the diagonals of the faces such that it doesn’t trespass a second time through the path made. Find the distance of the maximum journey this ant can make.


Let $ A_1,B_1,C_1 $ be the feet of the heights of an acute triangle $ ABC. $ On the segments $ B_1C_1,C_1A_1,A_1B_1, $ take the points $ X,Y, $ respectively, $ Z, $ such that
$ \left\{\begin{matrix}\frac{C_1X}{XB_1} =\frac{b\cos\angle BCA}{c\cos\angle ABC} \\  \frac{A_1Y}{YC_1} =\frac{c\cos\angle BAC}{a\cos\angle BCA} \\ \frac{B_1Z}{ZA_1} =\frac{a\cos\angle ABC}{b\cos\angle BAC} \end{matrix}\right. . $ Show that $ AX,BY,CZ, $ are concurrent.

Let $ ABC $ be a triangle in which $ O,I, $ are the circumcenter, respectively, incenter. The mediators of $ IA,IB,IC, $ form a triangle $ A_1B_1C_1. $ Show that $ \overrightarrow{OI}=\overrightarrow{OA_1} +\overrightarrow{OA_2} +\overrightarrow{OA_3} . $

Let $ABCD$ be a rectangle and the arbitrary points $E\in (CD)$ and $F \in (AD)$. The perpendicular from 
point $E$ on the line $FB$ intersects the line $BC$ at point $P$ and the perpendicular from point $F$ on the 
line $EB$ intersects the line $AB$ at point $Q$. Prove that the points $P, D$ and $Q$ are collinear.
Let $ABC$ be a triangle with $\angle A = 80^o$ and $\angle C = 30^o$. Consider the point $M$ inside the 
triangle $ABC$ so that $\angle MAC= 60^o$ and $\angle MCA = 20^o$. If $N$ is the intersection of the lines
$BM$ and $AC$ to show that a $MN$ is the bisector of the angle $\angle AMC$.
Let $ABCDA'B'C'D'$ be the rectangular parallelepiped. Let $M, N, P$ be midpoints of the edges $[AB], [BC],[BB']$ respectively . Let $\{O\} = A'N \cap C'M$. a) Prove that the points $D, O, P$ are collinear. b) Prove that $MC' \perp (A'PN)$ if and only if $ABCDA'B'C'D'$ is a cube.

Consider a right-angled triangle $ABC$, $\angle A = 90^{\circ}$ and points $D$ and $E$ on the leg $AB$ such that $\angle ACD \equiv \angle DCE \equiv \angle ECB$. Prove that if $3\overrightarrow{AD} = 2\overrightarrow{DE}$ and $\overrightarrow{CD} + \overrightarrow{CE} = 2\overrightarrow{CM}$ then $\overrightarrow{AB} = 4\overrightarrow{AM}$.

Let $AD$, $BE$, $CF$ be the heights of triangle $ABC$ and let $K$, $L$, $M$ be the orthocenters of triangles $AEF$, $BFD$ and $CDE$, respectively. Let $G_1$ and $G_2$ denote the centroids of triangles $DEF$ and $KLM$, respectively. Show that $HG_1 = G_1G_2$, where $H$ is the orthocenter of triangle $ABC$.

Consider $D$ the midpoint of the base $[BC]$ of the isosceles triangle ABC in which $\angle BAC < 90^o$. 
On the perpendicular from $B$ on the line $BC$ consider the point E such that $\angle EAB= \angle BAC$,  
and on the line passing though $C$ parallel to the line $AB$ we consider the point $F$ such that $F$ and $D$
are on different side of the line $AC$ and $\angle FAC = \angle CAD$. Prove that $AE = CF$ and $BF = EF$
Consider the isosceles right triangle$ ABC, \angle A = 90^o$, and point $D \in (AB)$ such that
$AD = \frac13 AB$. In the half-plane determined by the line $AB$ and point $C$ , consider a point
$E$ such that $\angle BDE = 60^o$ and $\angle DBE = 75^o$. Lines $BC$ and $DE$ intersect at
point $G$, and the line passing through point $G$ parallel to the line $AC$ intersects the line $BE$
at point $H$. Prove that the triangle $CEH$ is equilateral.

Let $ABCDA'B'C'D'$ be a rectangular parallelepiped and $M,N, P$ projections of points $A, C$ and
$B'$ respectively on the diagonal $BD'$.
a) Prove that $BM + BN + BP = BD'$.
b) Prove that   $3 (AM^2 + B'P^2 + CN^2)\ge 2D'B^2$ if and only if $ABCDA'B'C'D'$ is a cube.

Consider the rectangular parallelepiped $ABCDA'B'C'D' $ as such the measure of the dihedral angle
formed by the planes $(A'BD)$ and $(C'BD)$ is $90^o$ and the measure of the dihedral angle formed
by the planes $(AB'C)$ and $(D'B'C)$ is $60^o$. Determine and measure the dihedral angle formed
by the planes $(BC'D)$ and $(A'C'D)$.

Let $H$ be the orthocenter of the acute triangle $ABC.$ In the plane of the triangle $ABC$ we consider a point $X$ such that the triangle $XAH$ is right and isosceles, having the hypotenuse $AH,$ and $B$ and $X$ are on each part of the line $AH.$ Prove that $\overrightarrow{XA}+\overrightarrow{XC}+\overrightarrow{XH}=\overrightarrow{XB}$ if and only if $ \angle BAC=45^{\circ}.$

2019 Romania District X P3
Let $a,b,c$ be distinct complex numbers with $|a|=|b|=|c|=1.$ If $|a+b-c|^2+|b+c-a|^2+|c+a-b|^2=12,$ prove that the points of affixes $a,b,c$ are the vertices of an equilateral triangle.

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