MathDB

2

Part of 2011 Poland - Second Round

Problems(2)

Cyclic quadrilateral

Source: Polish MO second round 2011

2/19/2012
The convex quadrilateral ABCDABCD is given, AB<BCAB<BC and AD<CDAD<CD. P,QP,Q are points on BCBC and CDCD respectively such that PB=ABPB=AB and QD=ADQD=AD. MM is midpoint of PQPQ. We assume that ∠BMD=90∘\angle BMD=90^{\circ}, prove that ABCDABCD is cyclic.
geometrygeometric transformationreflectiongeometry unsolved
Maximal length of a sequence

Source: Polish MO second round 2011

2/19/2012
∀n∈Z+−{1,2}\forall n\in \mathbb{Z_{+}}-\{1,2\} find the maximal length of a sequence with elements from a set {1,2,…,n}\{1,2,\ldots,n\}, such that any two consecutive elements of this sequence are different and after removing all elements except for the four we do not receive a sequence in form x,y,x,yx,y,x,y (x≠yx\neq y).
combinatorics unsolvedcombinatorics