Found problems: 33
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.
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]
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.
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. $
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.
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. $
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. $
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 $
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. $
1987 Traian Lălescu, 1.4
Through a given point inside a circle, construct two perpendicular chords such that the sum of their lengths would be:
[b]a)[/b] maximum.
[b]b)[/b] minimum.
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. $
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 $10m + 10n + mn$.
2012 District Olympiad, 3
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:
[b]a)[/b] $ \frac{PM}{PN} =\frac{QB}{QC} . $
[b]b)[/b] The midpoints of the segments $ BM,CN,PQ $ are collinear.
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.
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]
2003 Alexandru Myller, 3
$ ABC $ and $ ADE $ are two triangles with $ \angle ABC=\angle ADE =90^{\circ } $ and such that $ AB=AD. $ The projection of $ B $ on $ AC $ is $ F, $ and the projection of $ D $ on $ AE $ is $ G. $ Prove that $ B,F,E $ are collinear if and only if $ D,G,C $ are collinear.
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. $
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.
1978 Romania Team Selection Test, 5
Find locus of points $ M $ inside an equilateral triangle $ ABC $ such that
$$ \angle MBC+\angle MCA +\angle MAB={\pi}/{2}. $$
1985 Traian Lălescu, 2.3
Let $ ABC $ a triangle, and $ P\neq B,C $ be a point situated upon the segment $ BC $ such that $ ABP $ and $ APC $ have the same perimeter. $ M $ represents the middle of $ BC, $ and $ I, $ the center of the incircle of $ ABC. $
Prove that $ IM\parallel AP. $
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.
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]
1978 Romania Team Selection Test, 4
Diagonals $ AC $ and $ BD $ of a convex quadrilateral $ ABCD $ intersect a point $ O. $ Prove that if triangles $ OAB,OBC,OCD $ and $ ODA $ have the same perimeter, then $ ABCD $ is a rhombus. What happens if $ O $ is some other point inside the quadrilateral?
2016 District Olympiad, 3
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. $
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.