Found problems: 33
1985 Traian Lălescu, 1.4
Let $ ABCD $ be a convex quadrilateral, and $ P $ be a point that isn't found on any of the lines formed by the sides of the quadrilateral. Prove that the centers of mass of the triangles $ PAB, PBC, PCD $ and $ PDA, $ form a parallelogram, and calculate the legths of its sides in terms of its diagonals.
1978 Romania Team Selection Test, 5
Find locus of points $ M $ inside an equilateral triangle $ ABC $ such that
$$ \angle MBC+\angle MCA +\angle MAB={\pi}/{2}. $$
2016 District Olympiad, 4
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:
[b]a)[/b] $ EBD\sim ABC $
[b]b)[/b] $ FGH\equiv ABC $
1985 Traian Lălescu, 2.1
Let $ ABC $ be a triangle. The perpendicular in $ B $ of the bisector of the angle $ \angle ABC $ intersects the bisector of the angle $ \angle BAC $ in $ M. $ Show that $ MC $ is perpendicular to the bisector of $ \angle BCA. $
2011 Gheorghe Vranceanu, 1
Let be a triangle $ ABC $ that's not equilateral, nor right-angled. Let $ A',B',C' $ be the feet of the heights of $ A,B,C, $ respectively. Prove that the Euler's lines of the triangles $ AB'C',BC'A',CA'B' $ meet at one point on the Euler's circle of $ ABC. $
Bangladesh Mathematical Olympiad 2020 Final, #2
Consider rectangle $ABCD$.$ E$ is the mid-point of $AD$ and $F$ is the mid-point of $ED$. $CE$ cuts $AB$ in $G$ and $BF$ cuts $CD$ in $H$ point. We can write ratio of areas of $BCG$ and $BCH$ triangles as $\frac{m}{n}$. Find the value of $10m + 10n + mn$.
1986 Traian Lălescu, 1.4
On the sides $ BC, CA $ and $ AB $ (extremities excluded) of the triangle $ ABC, $ consider the arbitrary points $ P,Q,R $ and the circumcenters $ O_1,O_2,O_3 $ of $ AQR,BRP,CPQ. $ Show that $ O_1O_2O_3\sim ABC. $
2007 Alexandru Myller, 3
Let $ ABC $ be a right angle in $ A, $ and $ M $ be the mid of $ BC. $ On the perpendicular of $ AM $ through $ A $ choose a point $ D $ so that $ DM $ meets $ AB $ at a point, namely $ P. $ Let $ E $ be the projection of $ D $ on $ BC. $ Show that $ \angle BPM =\angle EAC. $
1978 Romania Team Selection Test, 1
In a convex quadrilateral $ ABCD, $ let $ A’,B’ $ be the orthogonal projections to $ CD $ of $ A, $ respectively, $ B. $
[b]a)[/b] Assuming that $ BB’\le AA’ $ and that the perimeter of $ ABCD $ is $ (AB+CD)\cdot BB’, $ is $ ABCD $ necessarily a trapezoid?
[b]b)[/b] The same question with the addition that $ \angle BAD $ is obtuse.
2015 District Olympiad, 4
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.
1985 Traian Lălescu, 1.3
Let $ H $ be the orthocenter of $ ABC $ and $ A',B',C', $ the symmetric points of $ A,B,C $ with respect to $ H. $ The intersection of the segments $ BC,CA, AB $ with the circles of diameter $ A'H,B'H, $ respectively, $ C'H, $ consists of $ 6 $ points. Prove that these are concyclic.
2003 Romania National Olympiad, 1
Find the locus of the points $ M $ that are situated on the plane where a rhombus $ ABCD $ lies, and satisfy:
$$ MA\cdot MC+MB\cdot MD=AB^2 $$
[i]Ovidiu Pop[/i]
2017 District Olympiad, 3
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. $
[b]a)[/b] Show that $ \angle BME =75^{\circ } . $
[b]b)[/b] If the bisector of $ \angle CBE $ intersects $ CD $ in $ N, $ show that $ BMN $ is equilateral.
1978 Romania Team Selection Test, 2
Points $ A’,B,C’ $ are arbitrarily taken on edges $ SA,SB, $ respectively, $ SC $ of a tetrahedron $ SABC. $ Plane forrmed by $ ABC $ intersects the plane $ \rho , $ formed by $ A’B’C’, $ in a line $ d. $ Prove that, meanwhile the plane $ \rho $ rotates around $ d, $ the lines $ AA’,BB’ $ and $ CC’ $ are, and remain concurrent. Find de locus of the respective intersections.
2011 N.N. Mihăileanu Individual, 4
Consider a triangle $ ABC $ having incenter $ I $ and inradius $ r. $ Let $ D $ be the tangency of $ ABC $ 's incircle with $ BC, $ and $ E $ on the line $ BC $ such that $ AE $ is perpendicular to $ BC, $ and $ M\neq E $ on the segment $ AE $ such that $ AM=r. $
[b]a)[/b] Give an idenity for $ \frac{BD}{DC} $ involving only the lengths of the sides of the triangle.
[b]b)[/b] Prove that $ AB \cdot \overrightarrow{IC} +BC\cdot \overrightarrow{IA} +CA\cdot \overrightarrow{IB} =0. $
[b]c)[/b] Show that $ MI $ passes through the middle of the side $ BC. $
[i]Cătălin Zârnă[/i]
2019 Romania National Olympiad, 1
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 $
2000 Romania National Olympiad, 4
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.
1985 Traian Lălescu, 2.2
We are given the line $ d, $ and a point $ A $ which is not on $ d. $ Two points $ B $ and $ C $ move on $ d $ such that the angle $ \angle BAC $ is constant. Prove that the circumcircle of $ ABC $ is tangent to a fixed circle.
1985 Traian Lălescu, 1.2
For the triangles of fixed perimeter, find the maximum value of the product of the radius of the incircle with the radius of the excircle.
2016 Romania National Olympiad, 1
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.
2017 Romania National Olympiad, 1
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.
2016 Romania National Olympiad, 3
[b]a)[/b] 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]b)[/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. $
2007 Stars of Mathematics, 3
Let $ ABC $ be a triangle and $ A_1,B_1,C_1 $ the projections of $ A,B,C $ on their opposite sides. Let $ A_2,A_3 $ be the projection of $ A_1 $ on $ AB, $ respectively on $ AC. B_2,B_3,C_2,C_3 $ are defined analogously. Moreover, $ A_4 $ is the intersection of $ B_2B_3 $ with $ C_2C_3; B_4, $ the intersection of $C_2C_3 $ with $ A_2A_3; C_4, $ the intersection of $ A_2A_3 $ with $ B_2B_3. $
Show that $ AA_4,BB_4 $ and $ CC_4 $ are concurrent.
2017 Romania National Olympiad, 2
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. $
1998 Junior Balkan Team Selection Tests - Romania, 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.
[i]Dan Brânzei[/i]