MathDB

3

Part of 1997 Ukraine National Mathematical Olympiad

Problems(4)

circumcenters

Source: Ukraine 1997

7/18/2009
Consider acute-angled triangles ABC ABC and APQ APQ, where P P and Q Q lie on the side BC. BC. Prove that the circumcenter of △ABC \triangle ABC is closer to line BC BC than the circumcenter of △APQ. \triangle APQ.
geometrycircumcirclegeometry unsolved
incircle

Source: Ukraine 1997 grade 9

7/18/2009
The incircle of a triangle ABC ABC is tangent to its sides AB,BC,CA AB,BC,CA at M,N,K, M,N,K, respectively. A line l l through the midpoint D D of AC AC is parallel to MN MN and intersects the lines BC BC and BD BD at T T and S S, respectively. Prove that TC\equal{}KD\equal{}AS.
geometrygeometry proposed
sequence

Source: Ukraine 1997 grade 10

7/21/2009
A sequence is defined by a_1\equal{}a_2\equal{}a_3\equal{}1 and a_{n\plus{}3}\equal{}\minus{}a_n\minus{}a_{n\plus{}1} for all n∈N n \in \mathbb{N}. Prove that this sequence is not bounded , i.e. that for every M∈R M \in \mathbb{R} there is an n n for which ∣an∣>M. |a_n|>M.
algebra proposedalgebra
points

Source: Ukraine 1997 grade 11

7/22/2009
In space are given 1997 1997 points. Let M M and m m be the largest and smallest distance between two of the given points. Prove that Mm>9 \frac{M}{m}>9.
geometry3D geometryspherecombinatorics proposedcombinatorics