MathDB

2

Part of 2007 Turkey MO (2nd round)

Problems(2)

Turkey NMO 2007 Problem 2, Colored Cells on 2007x2007 board

Source: Turkey NMO 2007 Problem 2

11/13/2010
Some unit squares of 2007×2007 2007\times 2007 square board are colored. Let (i,j) (i,j) be a unit square belonging to the ithith line and jthjth column and Si,j S_{i,j} be the set of all colored unit squares (x,y)(x,y) satisfying x≤i,y≤j x\leq i, y\leq j . At the first step in each colored unit square (i,j)(i,j) we write the number of colored unit squares in Si,j S_{i,j} . In each step, in each colored unit square (i,j)(i,j) we write the sum of all numbers written in Si,j S_{i,j} in the previous step. Prove that after finite number of steps, all numbers in the colored unit squares will be odd.
combinatorics unsolvedcombinatorics
Turkey NMO 2007 Problem 5, CY=2MK

Source: Turkey NMO 2007 Problem 5

9/27/2011
Let ABCABC be a triangle with ∠B=90\angle B=90. The incircle of ABCABC touches the side BCBC at DD. The incenters of triangles ABDABD and ADCADC are XX and ZZ , respectively. The lines XZXZ and ADAD are intersecting at the point KK. XZXZ and circumcircle of ABCABC are intersecting at UU and VV. Let MM be the midpoint of line segment [UV][UV] . ADAD intersects the circumcircle of ABCABC at YY other than AA. Prove that ∣CY∣=2∣MK∣|CY|=2|MK| .
geometryincentercircumcircleinradiusrectanglegeometry unsolved