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Romania 2000-19 VII+ 102p

geometry problems from Romanian Nationan Mathematical Olympiads - Final Round
with aops links in the names
 
problems collected inside aops

2000 - 2019, grades IX-XII complete 
grades VII-VIII   2000 is missing only
Let $ A,B $ be two points in a plane and let two numbers $ a,b\in (0,1) . $ For each point $ M $ that is not on the line $ AB $ consider $ P $ on the segment $ AM $ and $ N $ on $ BM $ (both excluding the extremities) such that $ BN=b\cdot BM $ and $ AP=a\cdot AM. $ Find the locus of the points $ M $ for which $ AN=BP. $
Let $ I $ be the center of the incircle of a triangle $ ABC. $ Shw that, if for any point $ M $ on the segment $ AB $ (extremities excluded) there exist two points $ N,P $ on $ BC, $ respectively, $ AC $ (both excluding the extremities) such that the center of mass of $ MNP $ coincides with $ I, $ then $ ABC $ is equilateral.
Let be a tetahedron $ ABCD, $ and $ E $ be the projection of $ D $ on the plane formed by $ ABC. $ If $ \mathcal{A}_{\mathcal{R}} $ denotes the area of the region $ \mathcal{R}, $ show that the following affirmations are equivalent:
a) $ C=E\vee CE\parallel AB $
b) $ M\in\overline{CD}\implies\mathcal{A}_{ABM}^2=\frac{CM^2}{CD^2}\cdot\mathcal{A}_{ABD}^2 +\left( 1-\frac{CM^2}{CD^2}\right)\cdot\mathcal{A}_{ABC}^2  $
We consider a right trapezoid $ABCD$, in which $AB||CD,AB>CD,AD\perp AB$ and $AD>CD$. The diagonals $AC$ and $BD$ intersect at $O$. The parallel through $O$ to $AB$ intersects $AD$ in $E$ and $BE$ intersects $CD$ in $F$. Prove that $CE\perp AF$ if and only if $AB\cdot CD=AD^2-CD^2$ .
Consider the acute angle $ABC$. On the half-line $BC$ we consider the distinct points $P$ and $Q$ whose projections onto the line $AB$ are the points $M$ and $N$. Knowing that $AP=AQ$ and $AM^2-AN^2=BN^2-BM^2$, find the angle $ABC$.

We consider the points $A,B,C,D$, not in the same plane, such that $AB\perp CD$ and $AB^2+CD^2=AD^2+BC^2$.
a) Prove that $AC\perp BD$.
b) Prove that if $CD<BC<BD$, then the angle between the planes $(ABC)$ and $(ADC)$ is greater than $60^{\circ}$.

In the cube $ABCDA'B'C'D'$, with side $a$, the plane $(AB'D')$ intersects the planes $(A'BC),(A'CD),(A'DB)$ after the lines $d_1,d_2$ and $d_3$ respectively.
a) Show that the lines $d_1,d_2,d_3$ intersect pairwise.
b) Determine the area of the triangle formed by these three lines.

Let $ABC$ be a triangle $(A=90^{\circ})$ and $D\in (AC)$ such that $BD$ is the bisector of $B$. Prove that $BC-BD=2AB$ if and only if $\frac{1}{BD}-\frac{1}{BC}=\frac{1}{2AB} $

In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality $\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma $

2002 Romanian NMO grade VII  P3
Let $ABCD$ be a trapezium and $AB$ and $CD$ be it's parallel edges. Find, with proof, the set of interior points $P$ of the trapezium which have the property that $P$ belongs to at least two lines each intersecting the segments $AB$ and $CD$ and each dividing the trapezium in two other trapezoids with equal areas.

a) An equilateral triangle of sides $a$ is given and a triangle $MNP$ is constructed under the following conditions: $P\in (AB),M\in (BC),N\in (AC)$, such that $MP\perp AB,NM\perp BC$ and $PN\perp AC$. Find the length of the segment $MP$.
b) Show that for any acute triangle $ABC$ one can find points $P\in (AB),M\in (BC),N\in (AC)$ such that $MP\perp AB,NM\perp BC$ and $PN\perp AC$.

Let $[ABCDEF]$ be a frustum of a regular pyramid. Let $G$ and $G'$ be the centroids of bases $ABC$ and $DEF$ respectively. It is known that $AB=36,DE=12$ and $GG'=35$.
a) Prove that the planes $(ABF),(BCD),(CAE)$ have a common point $P$, and the planes $(DEC),(EFA),(FDB)$ have a common point $P'$, both situated on $GG'$.
b) Find the length of the segment $[PP']$.

The right prism $[A_1A_2A_3\ldots A_nA_1'A_2'A_3'\ldots A_n'],n\in\mathbb{N},n\ge 3$, has a convex polygon as its base. It is known that $A_1A_2'\perp A_2A_3',A_2A_3'\perp A_3A_4',$ $\ldots A_{n-1}A_n'\perp A_nA_1', A_nA_1'\perp A_1A_2'$. Show that:
a) $n=3$;
b) the prism is regular.

Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$. It is known that the symmetric point of $M$with respect to the line $GI$ lies on $AC$. Find the measure of $\measuredangle B$.

2002 Romanian NMO grade X P1
Let $X,Y,Z,T$ be four points in the plane. The segments $[XY]$ and $[ZT]$ are said to be connected, if there is some point $O$ in the plane such that the triangles $OXY$ and $OZT$ are right-angled at $O$ and isosceles.
Let $ABCDEF$ be a convex hexagon  such that the pairs of segments $[AB],[CE],$ and $[BD],[EF]$ are connected. Show that the points $A,C,D$ and $F$ are the vertices of a parallelogram and $[BC]$ and $[AE]$ are connected.

2003 Romanian NMO grade VII  P2
Compute the maximum area of a triangle having a median of length 1 and a median of length 2.

In triangle $ ABC$, $ P$ is the midpoint of side $ BC$. Let $ M\in(AB)$, $ N\in(AC)$ be such that $ MN\parallel BC$ and $ \{Q\}$ be the common point of $ MP$ and $ BN$. The perpendicular from $ Q$ on $ AC$ intersects $ AC$ in $ R$ and the parallel from $ B$ to $ AC$ in $ T$. Prove that:
a) $ TP\parallel MR$;
b) $ \angle MRQ=\angle PRQ$.

In tetrahedron $ ABCD$, $ G_1,G_2$ and $ G_3$ are barycenters of the faces $ ACD,ABD$ and $ BCD$ respectively.
a) Prove that the straight lines $ BG_1,CG_2$ and $ AG_3$ are concurrent.
b) Knowing that $ AG_3 =8,BG_1= 12$ and $ CG_2 =20$ compute the maximum possible value of the volume of $ ABCD$.

2003 Romanian NMO grade IX P3 (also)
Prove that the midpoints of the altitudes of a triangle are collinear if and only if the triangle is right.

Let $OABC$ be a tetrahedron such that $OA \perp OB \perp OC \perp  OA$, $r$ be the radius of its inscribed sphere and $H$ be the orthocenter of triangle $ABC$. Prove that $OH \le r(\sqrt3 +1)$

Find the locus of the points $M$ from the plane of a rhombus $ABCD$ such that $MA\cdot MC+ MB\cdot MD = AB^2$

Let be five nonzero complex numbers having the same absolute value and such that zero is equal to their sum, which is equal to the sum of their squares. Prove that the affixes of these numbers in the complex plane form a regular pentagon.

2004 Romanian NMO grade VII  P1
On the sides $AB,AD$ of the rhombus $ABCD$ are the points $E,F$ such that $AE=DF$. The lines $BC,DE$ intersect at $P$ and $CD,BF$ intersect at $Q$. Prove that:
a) $\frac{PE}{PD} + \frac{QF}{QB} = 1$;
b) $P,A,Q$ are collinear.
Let $ABCD$ be an orthodiagonal trapezoid such that $\measuredangle A = 90^{\circ}$ and $AB$ is the larger base. The diagonals intersect at $O$, $\left( OE \right.$ is the bisector of $\measuredangle AOD$, $E \in \left( AD \right)$ and $EF \| AB$, $F \in \left( BC \right)$. Let $P,Q$ the intersections of the segment $EF$ with $AC,BD$. Prove that:
a) $EP=QF$,
b) $EF=AD$.

Let $ABCD A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a truncated regular pyramid in which $BC^{\prime}$ and $DA^{\prime}$ are perpendicular.
a) Prove that $\measuredangle \left( AB^{\prime},DA^{\prime} \right) = 60^{\circ}$;
b) If the projection of $B^{\prime}$ on $(ABC)$ is the center of the incircle of $ABC$, then prove that $d \left( CB^{\prime},AD^{\prime} \right) = \frac12 BC^{\prime}$.

Let $H$ be the orthocenter of the acute triangle $ABC$. Let $BB'$ and $CC'$ be altitudes of the triangle ($B^{\prime} \in AC$, $C^{\prime} \in AB$). A variable line $\ell$ passing through $H$ intersects the segments $[BC']$ and $[CB']$ in $M$ and $N$. The perpendicular lines of $\ell$ from
$M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$. Determine the locus of the midpoint of the segment $[ PQ]$.

Let $ABCD$ be a tetrahedron in which the opposite sides are equal and form  equal angles. Prove that it is regular.

Let $n \geq 3$ be an integer and $F$ be the focus of the parabola $y^2=2px$. A regular polygon $A_1 A_2 \ldots A_n$ has the center in $F$ and none of its vertices lie on $Ox$. $\left( FA_1 \right., \left( FA_2 \right., \ldots, \left( FA_n \right.$ intersect the parabola at $B_1,B_2,\ldots,B_n$.
Prove that $FB_1 + FB_2 + \ldots + FB_n > np .  $

Let $ABCD$ be a parallelogram. The interior angle bisector of $\angle ADC$ intersects the line $BC$ in $E$, and the perpendicular bisector of the side $AD$ intersects the line $DE$ in $M$. Let $F= AM \cap BC$. Prove that:
a) $DE=AF$;
b) $AD\cdot AB = DE\cdot DM$. 
Let $ABCD$ be a quadrilateral with $AB\parallel CD$ and $AC \perp BD$. Let $O$ be the intersection of $AC$ and $BD$. On the rays $(OA$ and $(OB$ we consider the points $M$ and $N$ respectively such that $\angle ANC = \angle BMD = 90^\circ$. We denote with $E$ the midpoint of the segment $MN$. Prove that 
a) $\triangle OMN \sim \triangle OBA$;
b) $OE \perp AB$. 
We consider a cube with sides of length 1. Prove that a tetrahedron with vertices in the set of the vertices of the cube has the volume $\dfrac 16$ if and only if 3 of the vertices of the tetrahedron are vertices on the same face of the cube.

Let the $ABCA'B'C'$ be a regular prism. The points $M$ and $N$ are the midpoints of the sides $BB'$, respectively $BC$, and the angle between the lines $AB'$ and $BC'$ is of $60^\circ$. Let $O$ and $P$ be the intersection of the lines $A'C$ and $AC'$, with respectively $B'C$ and $C'N$. 
a) Prove that $AC' \perp (OPM)$;
b) Find the measure of the angle between the line $AP$ and the plane $(OPM)$.

Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$. Define the points $E=AD \cap BC$ and $I = AC\cap BD$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$. 
by Virgil Nicula
The base $A_{1}A_{2}\ldots A_{n}$ of the pyramid $VA_{1}A_{2}\ldots A_{n}$ is a regular polygon. Prove that if \[\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1},\] then the pyramid is regular.

Let $ABC$ be a triangle and the points $M$ and $N$ on the sides $AB$ respectively $BC$, such that $2 \cdot \frac{CN}{BC} = \frac{AM}{AB}$. Let $P$ be a point on the line $AC$. Prove that the lines $MN$ and $NP$ are perpendicular if and only if $PN$ is the interior angle bisector of $\angle MPC$.

In the acute-angle triangle $ABC$ we have $\angle ACB = 45^\circ$. The points $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$, and $H$ is the orthocenter of the triangle. We consider the points $D$ and $E$ on the segments $AA_1$ and $BC$ such that $A_1D = A_1E = A_1B_1$. Prove that 
a) $A_1B_1 = \sqrt{ \frac{A_1B^2+A_1C^2}{2} }$;
b) $CH=DE$.

We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.

Let $ABCDA_1B_1C_1D_1$ be a cube and $P$ a variable point on the side $[AB]$. The perpendicular plane on $AB$ which passes through $P$ intersects the line $AC'$ in $Q$. Let $M$ and $N$ be the midpoints of the segments $A'P$ and $BQ$ respectively. 
a) Prove that the lines $MN$ and $BC'$ are perpendicular if and only if $P$ is the midpoint of $AB$.
b) Find the minimal value of the angle between the lines $MN$ and $BC'$.

Let $\displaystyle ABC$ and $\displaystyle DBC$ be isosceles triangle with the base $\displaystyle BC$. We know that $\displaystyle \measuredangle ABD = \frac{\pi}{2}$. Let $\displaystyle M$ be the midpoint of $\displaystyle BC$. The points $\displaystyle E,F,P$ are chosen such that $\displaystyle E \in (AB)$, $\displaystyle P \in (MC)$, $\displaystyle C \in (AF)$, and $\displaystyle \measuredangle BDE = \measuredangle ADP = \measuredangle CDF$. Prove that $\displaystyle P$ is the midpoint of $\displaystyle EF$ and $\displaystyle DP \perp EF$.

We have a quadrilateral $ABCD$ inscribed in a circle of radius $r$, for which there is a point $P$ on $CD$ such that $CB=BP=PA=AB$.
a) Prove that there are points $A,B,C,D,P$ which fulfill the above conditions.
b) Prove that $PD=r$.
by Virgil Nicula
Consider the triangle $ ABC$ with $ m(\angle BAC=90^\circ)$ and $ AC =2AB$. Let $ P$ and $ Q$ be the midpoints of $ AB$ and $ AC$,respectively. Let $ M$ and $ N$ be two points found on the side $ BC$ such that $ CM =BN = x$. It is also known that $ 2S[MNPQ]= S[ABC]$. Determine $ x$ in function of $ AB$.

Consider the triangle $ ABC$ with $ m(\angle BAC) =90^\circ$ and $ AB < AC$.Let a point $ D$ on the side $ AC$ such that: $ m(\angle ACB) = m(\angle DBA)$.Let $ E$ be a point on the side $ BC$ such that $ DE\perp BC$.It is known that $ BD + DE =AC$. Find the measures of the angles in the triangle $ ABC$.

a) In a triangle $ MNP$, the lenghts of the sides are less than $ 2$. Prove that the lenght of the altitude corresponding to the side $ MN$ is less than $ \sqrt {4 - \frac {MN^2}{4}}$.
b) In a tetrahedron $ ABCD$, at least $ 5$ edges have their lenghts less than $ 2$.Prove that the volume of the tetrahedron is less than $ 1$.

Let $ ABCD$ be a tetrahedron.Prove that if a point $ M$ in a space satisfies the relation:
\begin{align*} MA^2 + MB^2 + CD^2 &= MB^2 + MC^2 + DA^2 \\ &= MC^2 + MD^2 + AB^2 \\ &= MD^2 + MA^2 + BC^2 . \end{align*} then it is found on the common perpendicular of the lines $ AC$ and $ BD$.

Let $ABC$ be an acute angled triangle and point $M$ chosen differently from $A,B,C$. Prove that $M$ is the orthocenter of triangle $ABC$ if and only if $\frac{BC}{MA}\vec{MA}+\frac{CA}{MB}\vec{MB}+\frac{AB}{MC}\vec{MC}= \vec{0}$

2008 Romanian NMO grade VII P1
Let $ ABC$ be an acute angled triangle with $ \angle B > \angle C$. Let $ D$ be the foot of the altitude from $ A$ on $ BC$, and let $ E$ be the foot of the perpendicular from $ D$ on $ AC$. Let $ F$ be a point on the segment $ (DE)$. Show that the lines $ AF$ and $ BF$ are perpendicular if and only if $ EF\cdot DC= BD \cdot DE$.

Let $ ABCD$ be a rectangle with center $ O$, $ AB\neq BC$. The perpendicular from $ O$ to $ BD$ cuts the lines $ AB$ and $ BC$ in $ E$ and $ F$ respectively. Let $ M,N$ be the midpoints of the segments $ CD,AD$ respectively. Prove that $ FM \perp EN$.

A tetrahedron has the side lengths positive integers, such that the product of any two opposite sides equals 6. Prove that the tetrahedron is a regular triangular pyramid in which the lateral sides form an angle of at least 30 degrees with the base plane.

Let $ ABCDA'B'C'D'$ be a cube. On the sides $ (A'D')$, $ (A'B')$ and $ (A'A)$ we consider the points $ M_1$, $ N_1$ and $ P_1$ respectively. On the sides $ (CB)$, $ (CD)$ and $ (CC')$ we consider the points $ M_2$, $ N_2$ and $ P_2$ respectively. Let $ d_1$ be the distance between the lines $ M_1N_1$ and $ M_2N_2$, $ d_2$ be the distance between the lines $ N_1P_1$ and $ N_2P_2$, and $ d_3$ be the distance between the lines $ P_1M_1$ and $ P_2M_2$. Suppose that the distances $ d_1$, $ d_2$ and $ d_3$ are pairwise distinct. Prove that the lines $ M_1M_2$, $ N_1N_2$ and $ P_1P_2$ are concurrent.

On the sides of triangle $ ABC$ we consider points $ C_1,C_2 \in (AB), B_1,B_2 \in (AC), A_1,A_2 \in (BC)$ such that triangles $ A_1,B_1,C_1$ and $ A_2B_2C_2$ have a common centroid. 
Prove that sets $ [A_1,B_1]\cap [A_2B_2], [B_1C_1]\cap[B_2C_2], [C_1A_1]\cap [C_2A_2]$ are not empty.

Let $ ABC$ be a triangle and the points $ D\in (BC)$, $ E\in (CA)$, $ F\in (AB)$ such that $\frac {BD}{DC} = \frac {CE}{EA} = \frac {AF}{FB}.$ Prove that if the circumcenters of the triangles $ DEF$ and $ ABC$ coincide then $ ABC$ is equilateral.

Consider triangles $ABC$ and  $A_1B_1C_1$ with $AB = A_1B_1$,  $\angle BAC = \angle B_1A_1C_1  = 60 ^o$ and  $\angle ABC  + \angle A_1B_1C_1 = 180^o$. Show that $\frac {1} {AC} + \frac {1} {A_1C_1} = \frac {1} {AB}$.

Let $\vartriangle ABC$ be  sharp and $D$ a point inside the triangle  such that $\angle ADB - \angle ACB = 90 ^o$, and $AC \cdot BD = AD \cdot BC$. Calculate $\angle DAC$ ,  $\angle DBC$ , and $\frac {AB \cdot CD} {AC \cdot BD}$.

On both sides of the plane $\vartriangle ABC$ are considered the points $S$ and $P$ such that $SA = SB = SC$ and $PA \perp PB \perp PC$. Knowing that $V_ {PABC} = 2V_ {SABC}$, show that the line $SP$ passes through the center of gravity of $\vartriangle ABC$.

On the sides $ AB,AC $ of a triangle $ ABC, $ consider the points $ M, $ respectively, $ N $ such that $ M\neq A\neq N  $ and $ \frac{MB}{MA}\neq\frac{NC}{NA}. $ Show that the line $ MN $ passes through a point not dependent on $ M $ and $ N. $

2010 Romanian NMO grade VII P2
Let $ABCD$ be a rectangle of centre $O$, such that $\angle DAC=60^{\circ}$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$, and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

In the isosceles triangle $ABC$, with $AB=AC$, the angle bisector of $\angle B$ meets the side $AC$ at $B'$. Suppose that $BB'+B'A=BC$. Find the angles of the triangle $ABC$.

Let $VABCD$ be a regular pyramid, having the square base $ABCD$. Suppose that on the line $AC$ lies a point $M$ such that $VM=MB$ and $(VMB)\perp (VAB)$. Prove that $4AM=3AC$.

2010 Romanian NMO grade IX P1
In a triangle $ABC$ denote by $D,E,F$ the points where the angle bisectors of $\angle CAB,\angle ABC,\angle BCA$ respectively meet it's circumcircle.
a) Prove that the orthocenter of triangle $DEF$ coincides with the incentre of triangle $ABC$.
b) Prove that if $\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}=0$, then the triangle $ABC$ is equilateral.

Prove that there is a similarity between a triangle $ABC$ and the triangle having as sides the medians of the triangle $ABC$ if and only if the squares of the lengths of the sides of triangle $ABC$ form an arithmetic sequence.

On the exterior of a non-equilateral triangle $ABC$ consider the similar triangles $ABM,BCN$ and $CAP$, such that the triangle $MNP$ is equilateral. Find the angles of the triangles $ABM,BCN$ and $CAP$.

In the convex quadrilateral $ABCD$ we have that $\angle BCD = \angle ADC  \ge 90 ^o$. The bisectors of $\angle BAD$ and $\angle ABC$ intersect in $M$. Prove that if $M \in CD$, then $M$ is the middle of $CD$.

Consider $\vartriangle ABC$ where $\angle ABC= 60 ^o$. Points $M$ and $D$ are on the sides $(AC)$, respectively $(AB)$, such that  $\angle BCA = 2   \angle MBC$, and $BD = MC$. Determine $\angle DMB$.

Let $VABC$ be a regular triangular pyramid with base $ABC$, of center $O$. Points $I$ and $H$ are the center of the inscribed circle, respectively the orthocenter $\vartriangle VBC$. Knowing that $AH = 3 OI$, determine the measure of the angle between the lateral edge of the pyramid and the plane of the base.

Let $ ABC $ be a triangle, $ I_a $ be center of the $ A\text{-excircle}. $ This excircle intersects the lines $ AB, AC, $ at $ P, $ respectively, $ Q. $ The line $ PQ $ intersects the lines $ I_aB,I_aC $ at $ D, $ respectively, $ E. $ Let $ A_1 $ be the intersection of $ DC $ with $ BE, $ and define, analogously, $ B_1,C_1. $ Show that $ AA_1,BB_1,CC_1 $ are concurrent.
Let $P$ be a point inside the square $ABCD$ and $PA = 1$, $PB = \sqrt2$ and $PC =\sqrt3$.
a) Determine the length of segment $[PD]$.
b) Determine the angle $\angle APB$.

Let $ABC$ be a triangle with right $\angle  A$. Consider points $D \in (AC)$ and $E \in (BD)$ such that $\angle ABC = \angle ECD = \angle CED$. Prove that $BE = 2 \cdot AD$

2012 Romanian NMO grade VIII P3
Let $ACD$ and $BCD$ be acute-angled triangles located in different planes. Let $G$ and $H$ be the centroid and the orthocenter respectively of the $BCD$ triangle; Similarly let $G'$ and $H'$ be  the centroid and the orthocenter of the $ACD$ triangle. Knowing that  $HH'$ is perpendicular to the plane $(ACD)$,  show that  $GG' $ is perpendicular to the plane $(BCD)$.

The altitude $[BH]$ dropped onto the hypotenuse of a triangle $ABC$ intersects the bisectors $[AD]$ and $[CE]$ at $Q$ and $P$ respectively. Prove that the line passing through the midpoints of the segments $[QD]$ and $[PE]$ is parallel to the line $AC$ .

Let $M=\{x\in\mathbb{C}\, |\, |z|=1,\ \text{Re}\, z\in\mathbb{Q}\}\, .$ Prove that there exist infinitely many equilateral triangles in the complex plane having all affixes of their vertices in the set $M$ .

In the triangle  $ABC$, the angle - bisector $AD$ ($D \in BC$) and the median $BE$ ($E \in AC$) intersect at point $P$. Lines $AB$ and $CP$ intesect at point $F$. The parallel through $B$ to $CF$ intersects $DF$ at point $M$. Prove that $DM = BF$

Let $ABCD$ be a rectangle with $5AD <2 AB$ . On the  side $AB$ consider the points $S$ and $T$ such that $AS = ST = TB$.  Let $M, N$ and $P$ be the projections of points $A, S$ and $T$ on lines $DS, DT$ and $DB$ respectively .Prove that the points  $M, N$, and $P$ are collinear if and only if $15 AD^2 = 2 AB^2$
Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$
a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$
b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$

Outside the square $ABCD$,  the rhombus $BCMN$  is constructed with angle $BCM$ obtuse . Let $P$ be the point of intersection of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and  the triangle $DPM$ is right isosceles .
Outside the square $ABCD$ is constructed the right isosceles triangle $ABD$ with hypotenuse $[AB]$. Let $N$ be the midpoint of the side $[AD]$ and ${M} = CE \cap  AB$, ${P} = CN \cap AB$ , ${F} = PE \cap MN$. On the line $FP$ the point $Q$ is considered such that the $[CE$ is the bisector of the angle $QCB$. Prove that  $MQ \perp CF$.

Let $ABCDA'B'C'D'$ be a cube with side $AB = a$. Consider points $E \in (AB)$ and $F \in (BC)$ such that $AE + CF = EF$.
a) Determine the measure the angle formed by the planes $(D'DE)$ and $(D'DF)$.
b) Calculate the distance from $D'$ to the line $EF$.

2014 Romanian NMO grade IX P3
Let $ P,Q $ be the midpoints of the diagonals $ BD, $ respectively, $ AC, $ of the quadrilateral $ ABCD, $ and points $ M,N,R,S $ on the segments $ BC,CD,PQ, $ respectively $ AC, $ except their extremities, such that $ \frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} . $ Show that the center of mass of the triangle $ AMN $ is situated on the segment $ RS. $

Let $ ABCD $ be a quadrilateral inscribed in a circle of diameter $ AC. $ Fix points $ E,F $ of segments $ CD, $ respectively, $ BC $ such that $ AE $ is perpendicular to $ DF $ and $ AF $ is perpendicular to $ BE. $ Show that $ AB=AD. $

Consider the triangle $ABC$ with $\angle = 90^o, AC <AB$. On the rays $(BA$ and $(AC$, consider points $E$ and $D$ respectively such that $A \in  (BE), C  \in (AD), AE = AC$ and $AD = AB$. Let $M , N$ be the midpoints of the segments $[BC]$ and $[DE]$ respectively, and $\{R\} = EC \cap BD$. Prove that $MN = RA$.

Consider the triangle $ABC, \angle B = 90^o$. The circle inscribed in the triangle $ABC$ has center $I$, and the points $F, D ,E$ are the points of contact of this circle with the sides $[AB], [BC],[AC]$ respectively . If $CI \cap EF = \{M\} $and $DM \cap AB = \{N\}$, show that:
a) $AI = ND$
b)$ FM =\frac{EI \cdot EM}{EC}$

Let $ABCA'B'C'$ be a trunk of a triangular pyramid. Consider points $D \in (AA'), E \in (BB')$ and $F \in  (CC')$ so that the planes $(AEF)$ and $(DB'C')$ are parallel. Prove that the planes $(A'EF)$ and $(DBC)$ are parallel.

Let be a point $ P $ in the interior of a triangle $ ABC. $ The lines $ AP,BP,CP $ meet $ BC,AC, $ respectively, $ AB $ at $ A_1,B_1, $ respectively, $ C_1. $ If  $ \mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} , $ show that $ P $ lies on a median of $ ABC. $

$ \mathcal{A} $ denotes area.
2016 Romanian NMO grade VII P2
Consider the triangle $ABC$, where $\angle B= 30^o, \angle C  = 15^o$, and $M$ is the midpoint of the side $[BC]$. Let point $N \in (BC)$ be such that $[NC] = [AB]$. Show that $[AN$ is the angle bisector of $MAC$ 

Consider the isosceles right triangle $ABC$, with $\angle A = 90^o$ and the point $M \in (BC)$ such that $\angle AMB  = 75^o$. On the inner bisector of the  angle $MAC$ take a point $F$ such that $BF = AB$. Prove that:
a) the lines $AM$ and $BF$ are perpendicular.
b) the triangle $CFM$ is isosceles.

In a cube $ABCDA'B'C'D' $two points are considered, $M \in (CD')$ and $N \in (DA')$. Show that the $MN$ is common perpendicular to the lines $CD'$ and $DA'$ if  and only if  $\frac{D'M}{D'C}=\frac{DN}{DA'} =\frac{1}{3}$

2016 Romanian NMO grade IX P1
The orthocenter $ H $ of a triangle $ ABC $ is distinct from its vertices and its circumcenter $ O. $ $ M,N,P $ are the circumcenters of the triangles $ HBC,HCA, $ respectively, $ HAB. $ Prove that $ AM,BN,CP $ and $ OH $ are concurrent.

a) Let be two nonzero complex numbers $ a,b. $ Show that the area of the triangle formed by the representations of the affixes $ 0,a,b $ in the complex plane is $ \frac{1}{4}\left| \overline{a} b-a\overline{b} \right| . $
b) Let be an equilateral triangle $ ABC, $ its circumcircle $ \mathcal{C} , $ its circumcenter $ O, $ and two distinct points $ P_1,P_2 $ in the interior of $ \mathcal{C} . $ Prove that we can form two triangles with sides $ P_1A,P_1B,P_1C, $ respectively, $ P_2A,P_2B,P_2C, $ whose areas are equal if and only if $ OP_1=OP_2. $

2017 Romanian NMO grade VII P2
Consider the triangle $ABC$, with $\angle A= 90^o, \angle B = 30^o$, and $D$ is the foot of the altitude from $A$. Let the point $E \in  (AD)$ such that $DE = 3AE$ and $F$ the foot of the perpendicular from $D$ to the line $BE$.
a) Prove that $AF \perp FC$.

b) Determine the measure of the angle $AFB$.

In the square $ABCD$ denote by $M$ the midpoint of the side $[AB]$,  with $P$ the projection of point $B$ on the line $CM$ and with $N$ the midpoint of the segment $[CP]$, Bisector of the angle $DAN$ intersects the line $DP$  at point $Q$. Show that the quadrilateral $BMQN$ is a parallelogram.

Prove the following:
a) If $ABCA'B'C'$ is a right prism and $M \in (BC), N \in (CA), P \in (AB)$ such that  $A'M, B'N$ and $C'P$ are perpendicular each other  and concurrent, then the prism $ABCA'B'C'$ is regular.
b) If $ABCA'B'C'$ is a regular prism and $\frac{AA'}{AB}=\frac{\sqrt6}{4}$ , then there are $M \in (BC), N \in (CA), P \in (AB)$  so that the lines $A'M, B'N$ and $C'P$ are perpendicular each other and concurrent.

2017 Romanian NMO grade IX P1
Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.

Let be a square $ ABCD, $ a point $ E $ on $ AB, $ a point $ N $ on $ CD, $ points $ F,M $ on $ BC, $ name $ P $ the intersection of $ AN $ with $ DE, $ and name $ Q $ the intersection of $ AM $ with $ EF. $ If the triangles $ AMN $ and $ DEF $ are equilateral, prove that $ PQ=FM. $

2018 Romanian NMO grade VII P2
In the square $ABCD$ the point $E$ is located on the side $[AB]$, and $F$ is the foot of the perpendicular from $B$ on the line $DE$. The point $L$ belongs to the line $DE$, such that  $F$ is between $E$ and $L$, and $FL = BF$. $N$ and $P$ are symmetric of the points $A , F$ with respect to the lines $DE, BL$, respectively. Prove that:
a) The quadrilateral $BFLP$ is square and the quadrilateral $ALND$ is rhombus.

b) The area of ​​the rhombus $ALND$ is equal to the difference between the areas of the squares $ABCD$ and $BFLP$.

On the sides $[AB]$ and $[BC]$ of the parallelogram $ABCD$ are constructed the equilateral triangles $ABE$ and $BCF,$ so that the points $D$ and $E$ are on the same side  of the line $AB$, and $F$ and $D$ on different sides of the line $BC$. If the points $E,D$ and $F$ are collinear, then prove that $ABCD$ is rhombus.
In the rectangular parallelepiped $ABCDA'B'C'D'$ we denote by $M$ the center of the face $ABB'A'$. We denote by $M_1$ and $M_2$ the projections of $M$ on the lines $B'C$ and $AD'$ respectively. Prove that:
a)  $MM_1 =  MM_2$ 
b) if $(MM_1M_2) \cap (ABC) = d$, then $d \parallel  AD$;
c)  $\angle (MM_1M_2), (A B C)= 45^ o \Leftrightarrow  \frac{BC}{AB}=\frac{BB'}{BC}+\frac{BC}{BB'}$.

2018 Romanian NMO grade IX P1
Prove that if in a triangle the orthocenter, the centroid and the incenter are collinear, then the triangle is isosceles.

Let $ABC$ be a triangle, $O$ its circumcenter and $R=1$ its circumradius. Let $G_1,G_2,G_3$ be the centroids of the triangles $OBC, OAC$ and $OAB.$ Prove that the triangle $ABC$ is equilateral if and only if $$AG_1+BG_2+CG_3=4$$

Let $ABCD$ be a square and $E$ a point on the side $(CD)$. Squares $ENMA$ and $EBQP$ are constructed outside the triangle $ABE$. Prove that:
a) $ND = PC$
b) $ND\perp  PC$.

Let $ABC$ be a triangle in which $\angle ABC = 45^o$ and  $\angle  BAC > 90^o$. Let $O$ be the midpoint of the side $[BC]$. Consider the point $M \in  (AC)$ such that $\angle COM =\angle CAB$. Perpendicular from $M$ on $AC$ intersects line $AB$ at point $P$.
a) Find the  measure of the angle $\angle  BCP$.
b) Show that if $\angle BAC  = 105^o$, then $PB = 2MO$.

In the regular hexagonal prism $ABCDEFA_1B_1C_1D_1E_1F_1$, We construct $, Q$, the projections of point $A$ on the lines $A_1B$ and $A_1C$ repsectilvely. We construct $R,S$, the projections of point  $D_1$ on the lines $A_1D$ and $C_1D$ respectively.
a) Determine the measure of the angle between the planes $(AQP)$ and $(D_1RS)$.
b) Show that $\angle AQP = \angle D_1RS$.

2019 Romanian NMO grade IX P1
Let be a point $ P $ in the interior of a triangle $ ABC $ such that $ BP=AC, M $ be the middlepoint of the segment $ AP,   R $ be the middlepoint of $ BC $ and $ E $ be the intersection of $ BP $ with $ AC. $ Prove that the bisector of $ \angle BEA $ is perpendicular on $ MR $

2019 Romanian NMO grade IX P2
Find all natural numbers which are the cardinal of a set of nonzero Euclidean vectors whose sum is $ 0, $ the sum of any two of them is nonzero, and their magnitudes are equal.

2019 Romanian NMO grade X P1
Find the number of trapeziums that it can be formed with the vertices of a regular polygon.



source: forum.gil.ro

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