MathDB

2

Part of 2011 Czech-Polish-Slovak Match

Problems(2)

Changing numbers on a blackboard to zeroes

Source: Czech-Polish-Slovak Match, 2011

8/9/2011
Written on a blackboard are nn nonnegative integers whose greatest common divisor is 11. A move consists of erasing two numbers xx and yy, where x≥yx\ge y, on the blackboard and replacing them with the numbers x−yx-y and 2y2y. Determine for which original nn-tuples of numbers on the blackboard is it possible to reach a point, after some number of moves, where n−1n-1 of the numbers of the blackboard are zeroes.
invariantnumber theorygreatest common divisorinductioncombinatorics unsolvedcombinatorics
Concurrent lines through midpoints in a quadrilateral

Source: Czech-Polish-Slovak Match, 2011

8/9/2011
In convex quadrilateral ABCDABCD, let MM and NN denote the midpoints of sides ADAD and BCBC, respectively. On sides ABAB and CDCD are points KK and LL, respectively, such that ∠MKA=∠NLC\angle MKA=\angle NLC. Prove that if lines BDBD, KMKM, and LNLN are concurrent, then ∠KMN=∠BDCand∠LNM=∠ABD. \angle KMN = \angle BDC\qquad\text{and}\qquad\angle LNM=\angle ABD.
symmetrygeometrycyclic quadrilateralgeometry unsolved