MathDB

3

Part of 2002 District Olympiad

Problems(6)

polyhedron is a regular pyramid, angle between line and plane

Source: 2002 Romania District VIII P3

5/24/2020
Consider the regular pyramid VABCDVABCD with the vertex in VV which measures the angle formed by two opposite lateral edges is 45o45^o. The points M,N,PM,N,P are respectively, the projections of the point AA on the line VCVC, the symmetric of the point MM with respect to the plane (VBD)(VBD) and the symmetric of the point NN with respect to OO. (OO is the center of the base of the pyramid.)
a) Show that the polyhedron MDNBPMDNBP is a regular pyramid.
b) Determine the measure of the angle between the line NDND and the plane (ABC)(ABC)
geometry3D geometrypyramidpolyhedronangles
mipoint triangle is an equilateral also (2002 Romania District VII P3)

Source:

5/23/2020
Consider the equilateral triangle ABCABC with center of gravity GG. Let MM be a point, inside the triangle and OO be the midpoint of the segment MGMG. Three segments go through MM, each parallel to one side of the triangle and with the ends on the other two sides of the given triangle.
a) Show that OO 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.
geometrymidpointEquilateralCentroiddistance
centers of mass, and a geometric double inequality

Source: Romanian District Olympiad 2002, Grade IX, Problem 3

10/7/2018
Let G G be the center of mass of a triangle ABC, ABC, and the points M,N,P M,N,P on the segments AB,BC, AB,BC, respectively, CA CA (excluding the extremities) such that AMMB=BNNC=CPPA. \frac{AM}{MB} =\frac{BN}{NC} =\frac{CP}{PA} . G1,G2,G3 G_1,G_2,G_3 are the centers of mass of the triangles AMP,BMN, AMP, BMN, respectively, CNP. CNP. Pove that:
a) The centers of mas of ABC ABC and G1G2G3 G_1G_2G_3 are the same. b) For any planar point D, D, the inequality 3⋅DG<DG1+DG2+DG3<DA+DB+DC 3\cdot DG< DG_1+DG_2+DG_3<DA+DB+DC holds.
inequalitiesgeometrycenter of mass
Romania District Olympiad 2002 - Grade XI

Source:

3/18/2011
a)Find a matrix A∈M3(C)A\in \mathcal{M}_3(\mathbb{C}) such that A2≠O3A^2\neq O_3 and A3=O3A^3=O_3. b)Let n,p∈{2,3}n,p\in\{2,3\}. Prove that if there is bijective function f:Mn(C)→Mp(C)f:\mathcal{M}_n(\mathbb{C})\rightarrow \mathcal{M}_p(\mathbb{C}) such that f(XY)=f(X)f(Y), ∀X,Y∈Mn(C)f(XY)=f(X)f(Y),\ \forall X,Y\in \mathcal{M}_n(\mathbb{C}), then n=pn=p.
Ion Savu
linear algebramatrixfunctionlinear algebra unsolved
Integrand of recurrent type (1+x)^r.ln^s x

Source: Romanian District Olympiad 2002, Grade XII, Problem 3

10/7/2018
a) Calculate lim⁡n→∞∫0αln⁡(1+x+x2+⋯+xn−1)dx, \lim_{n\to\infty} \int_0^{\alpha } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx , for all α∈(0,1). \alpha\in (0,1) .
b) Calculate lim⁡n→∞∫01ln⁡(1+x+x2+⋯+xn−1)dx. \lim_{n\to\infty} \int_0^{1 } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx .
logarithmscalculusintegrationreal analysis
logarithmic inequality involving only primes

Source: Romanian District Olympiad 2002, Grade X, Problem 3

10/7/2018
Let be two real numbers a,b, a,b, that satisfy 3a+13b=17a 3^a+13^b=17^a and 5a+7b=11b. 5^a+7^b=11^b. Show that a<b. a<b.
number theorylogarithmsinequalities