MathDB

3

Part of 2005 District Olympiad

Problems(6)

AM, GM algebraic computations

Source:

3/5/2005
We denote with mam_a, mgm_g the arithmetic mean and geometrical mean, respectively, of the positive numbers x,yx,y. a) If ma+mg=y−xm_a+m_g=y-x, determine the value of xy\dfrac xy; b) Prove that there exists exactly one pair of different positive integers (x,y)(x,y) for which ma+mg=40m_a+m_g=40.
equifacial tetrahedron and circumcircles of faces

Source: RMO District 2005, 8th Grade, Problem 3

3/5/2005
Prove that if the circumcircles of the faces of a tetrahedron ABCDABCD have equal radii, then AB=CDAB=CD, AC=BDAC=BD and AD=BCAD=BC.
geometry3D geometrytetrahedroncircumcircletrigonometrytrig identitiesLaw of Sines
about midpoints and centroids

Source: RMO District 2005, 9th Grade, Problem 3

3/5/2005
Let ABCABC be a non-right-angled triangle and let HH be its orthocenter. Let M1,M2,M3M_1,M_2,M_3 be the midpoints of the sides BCBC, CACA, ABAB respectively. Let A1A_1, B1B_1, C1C_1 be the symmetrical points of HH with respect to M1M_1, M2M_2 and M3M_3 respectively, and let A2A_2, B2B_2, C2C_2 be the orthocenters of the triangles BA1CBA_1C, CB1ACB_1A and AC1BAC_1B respectively. Prove that: a) triangles ABCABC and A2B2C2A_2B_2C_2 have the same centroid; b) the centroids of the triangles AA1A2AA_1A_2, BB1B2BB_1B_2, CC1C2CC_1C_2 form a triangle similar with ABCABC.
geometrygeometric transformationhomothetygeometry proposed
another solid geometry problem

Source: RMO District 2005, 10th Grade, Problem 3

3/5/2005
Let OO be a point equally distanced from the vertices of the tetrahedron ABCDABCD. If the distances from OO to the planes (BCD)(BCD), (ACD)(ACD), (ABD)(ABD) and (ABC)(ABC) are equal, prove that the sum of the distances from a point M∈int[ABCD]M \in \textrm{int}[ABCD], to the four planes, is constant.
geometry3D geometrytetrahedroncircumcirclegeometry proposed
Romania District Olympiad 2005 - Grade XI

Source:

4/10/2011
a)Let A,B∈M3(R)A,B\in \mathcal{M}_3(\mathbb{R}) such that rank A>rank B\text{rank}\ A>\text{rank}\ B. Prove that rank A2≥rank B2\text{rank}\ A^2\ge \text{rank}\ B^2.
b)Find the non-constant polynomials f∈R[X]f\in \mathbb{R}[X] such that (∀)A,B∈M4(R)(\forall)A,B\in \mathcal{M}_4(\mathbb{R}) with rank A>rank B\text{rank}\ A>\text{rank}\ B, we have rank f(A)>rank f(B)\text{rank}\ f(A)>\text{rank}\ f(B).
algebrapolynomiallinear algebralinear algebra unsolved
finite number of finite order elements in a group

Source: RMO District 2005, 12th Grade, Problem 3

3/5/2005
Let (G,⋅)(G,\cdot) be a group and let FF be the set of elements in the group GG of finite order. Prove that if FF is finite, then there exists a positive integer nn such that for all x∈Gx\in G and for all y∈Fy\in F, we have xny=yxn. x^n y = yx^n.
superior algebrasuperior algebra solved