MathDB

2

Part of 2010 International Zhautykov Olympiad

Problems(2)

IZhO 2010 1-st round-2

Source:

9/4/2010
In a cyclic quadrilateral ABCDABCD with AB=ADAB=AD points MM,NN lie on the sides BCBC and CDCD respectively so that MN=BM+DNMN=BM+DN . Lines AMAM and ANAN meet the circumcircle of ABCDABCD again at points PP and QQ respectively. Prove that the orthocenter of the triangle APQAPQ lies on the segment MNMN .
geometrycircumcirclecyclic quadrilateralgeometry proposed
IZhO 2010 2-nd round-2

Source:

9/4/2010
In every vertex of a regular nn -gon exactly one chip is placed. At each stepstep one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by [n2][\frac{n}{2}] positions clockwise from its initial position.
number theorygreatest common divisormodular arithmeticcombinatorics proposedcombinatorics