MathDB

4

Part of 2008 Romania National Olympiad

Problems(6)

Triangle ABC

Source: Romania NMO 2008, 9 form, Problem 4

4/30/2008
On the sides of triangle ABC ABC we consider points C1,C2∈(AB),B1,B2∈(AC),A1,A2∈(BC) C_1,C_2 \in (AB), B_1,B_2 \in (AC), A_1,A_2 \in (BC) such that triangles A1,B1,C1 A_1,B_1,C_1 and A2B2C2 A_2B_2C_2 have a common centroid. Prove that sets [A1,B1]∩[A2B2],[B1C1]∩[B2C2],[C1A1]∩[C2A2] [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.
geometry proposedgeometry
A rectangle of center O

Source: RMO 2008, Grade 7, Problem 4

4/30/2008
Let ABCD ABCD be a rectangle with center O O, AB≠BC AB\neq BC. The perpendicular from O O to BD BD cuts the lines AB AB and BC BC in E E and F F respectively. Let M,N M,N be the midpoints of the segments CD,AD CD,AD respectively. Prove that FM⊥EN FM \perp EN.
geometryrectangletrigonometrygeometric transformationrotationhomothetyanalytic geometry
The cube

Source: RMO 2008, Grade 8, Problem 4

4/30/2008
Let ABCDA′B′C′D′ ABCDA'B'C'D' be a cube. On the sides (A′D′) (A'D'), (A′B′) (A'B') and (A′A) (A'A) we consider the points M1 M_1, N1 N_1 and P1 P_1 respectively. On the sides (CB) (CB), (CD) (CD) and (CC′) (CC') we consider the points M2 M_2, N2 N_2 and P2 P_2 respectively. Let d1 d_1 be the distance between the lines M1N1 M_1N_1 and M2N2 M_2N_2, d2 d_2 be the distance between the lines N1P1 N_1P_1 and N2P2 N_2P_2, and d3 d_3 be the distance between the lines P1M1 P_1M_1 and P2M2 P_2M_2. Suppose that the distances d1 d_1, d2 d_2 and d3 d_3 are pairwise distinct. Prove that the lines M1M2 M_1M_2, N1N2 N_1N_2 and P1P2 P_1P_2 are concurrent.
geometry3D geometry
Infinite sets

Source: RMO 2008, Grade 10, Problem 4

4/30/2008
We consider the proposition p(n) p(n): n^2\plus{}1 divides n! n!, for positive integers n n. Prove that there are infinite values of n n for which p(n) p(n) is true, and infinite values of n n for which p(n) p(n) is false.
number theorynumber theory proposed
Antisymetric matrix

Source: RMO 2008, 11th Grade, Problem 4

4/30/2008
Let A\equal{}(a_{ij})_{1\leq i,j\leq n} be a real n×n n\times n matrix, such that a_{ij} \plus{} a_{ji} \equal{} 0, for all i,j i,j. Prove that for all non-negative real numbers x,y x,y we have \det(A\plus{}xI_n)\cdot \det(A\plus{}yI_n) \geq \det (A\plus{}\sqrt{xy}I_n)^2.
linear algebramatrixalgebrapolynomialinequalitieslinear algebra unsolved
Endomorphisms on finite groups

Source: RMO 2008, Grade 12, Problem 4

4/30/2008
Let G \mathcal G be the set of all finite groups with at least two elements. a) Prove that if G∈G G\in \mathcal G, then the number of morphisms f:G→G f: G\to G is at most nnp \sqrt [p]{n^n}, where p p is the largest prime divisor of n n, and n n is the number of elements in G G. b) Find all the groups in G \mathcal G for which the inequality at point a) is an equality.
inequalitiesgroup theoryabstract algebravectorsuperior algebrasuperior algebra unsolved