3
Part of 2006 Tuymaada Olympiad
Problems(2)
a line and a fixed point
Source: tuymaada 2006 - problem 3
7/17/2006
A line is given in the plane. Let and another point, not on , and such that is not perpendicular on . Let be a variable circle touching at and letting outside, and and the points on such that and are tangent to the circle. Prove that the line passes through a fixed point.Proposed by F. Bakharev
geometrygeometric transformationhomothetypower of a pointradical axisgeometry unsolved
colouring squares
Source: tuymaada 2006 - problem 7
7/17/2006
From a rectangle divided into unit squares, we cut the corner, which consists of the first row and the first column. (that is, the corner has unit squares). For the following, when we say corner we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of .Proposed by S. Berlov
geometryrectanglegeometric transformationrotationinductionvectorabsolute value