MathDB

3

Part of 2009 Junior Balkan Team Selection Test

Problems(2)

Minimum number of fields with figures

Source: Serbian JBTST 3, Day 1

5/30/2009
On each field of the board n×n n\times n there is one figure, where n≥2n\ge 2. In one move we move every figure on one of its diagonally adjacent fields. After one move on one field there can be more than one figure. Find the least number of fields on which there can be all figures after some number of moves.
combinatorics proposedcombinatorics
Proving a kite

Source: Serbian JBTST 3, Day 2

5/31/2009
Let ABCD ABCD be a convex quadrilateral, such that \angle CBD\equal{}2\cdot\angle ADB, \angle ABD\equal{}2\cdot\angle CDB and AB\equal{}CB. Prove that quadrilateral ABCD ABCD is a kite.
geometryparallelogramperpendicular bisectorgeometry proposed