MathDB

3

Part of 1998 Turkey MO (2nd round)

Problems(2)

Colored Points on a Circle

Source: Turkish NMO 1998, 3. Problem

7/31/2011
The points of a circle are colored by three colors. Prove that there exist infinitely many isosceles triangles inscribed in the circle whose vertices are of the same color.
Ramsey Theorycombinatorics proposedcombinatorics
Colored Points on a nxn Chessboard, finding a limit

Source: Turkish NMO 1998, 6. Problem

7/31/2011
Some of the vertices of unit squares of an n×nn\times n chessboard are colored so that any k×kk\times k ( 1≤k≤n1\le k\le n) square consisting of these unit squares has a colored point on at least one of its sides. Let l(n)l(n) denote the minimum number of colored points required to satisfy this condition. Prove that \underset{n\to \infty }{\mathop \lim }\,\frac{l(n)}{{{n}^{2}}}=\frac{2}{7}.
limitcombinatorics proposedcombinatorics