MathDB

3

Part of 2004 Tuymaada Olympiad

Problems(2)

Find the circumradius of some triangle.

Source: Tuymaada 2004, day 1, problem 3. - Authors : F. Bakharev, F. Petrov.

5/25/2007
An acute triangle ABCABC is inscribed in a circle of radius 1 with centre O;O; all the angles of ABCABC are greater than 45∘.45^\circ. B1B_{1} is the foot of perpendicular from BB to CO,CO, B2B_{2} is the foot of perpendicular from B1B_{1} to AC.AC. Similarly, C1C_{1} is the foot of perpendicular from CC to BO,BO, C2C_{2} is the foot of perpendicular from C1C_{1} to AB.AB. The lines B1B2B_{1}B_{2} and C1C2C_{1}C_{2} intersect at A3.A_{3}. The points B3B_{3} and C3C_{3} are defined in the same way. Find the circumradius of triangle A3B3C3.A_{3}B_{3}C_{3}.
Proposed by F.Bakharev, F.Petrov
geometrycircumcircleparallelogramcongruent trianglesgeometry proposed
Arrangement of an n x n table with entries in {0, 1}.

Source: Tuymaada 2004, day 2, problem 3. - Author : O. Vanyushina.

5/25/2007
Zeroes and ones are arranged in all the squares of n×nn\times n table. All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form [asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy] (consisting of a square and its neighbours from left and from below) is even. Prove that no two rows of the table are identical.
Proposed by O. Vanyushina
vectorfunctioninductioncombinatorics proposedcombinatorics