MathDB

2

Part of 1993 Vietnam National Olympiad

Problems(2)

locus of center parallelogram

Source: 31-th Vietnamese Mathematical Olympiad 1993

2/17/2007
ABCDABCD is a quadrilateral such that ABAB is not parallel to CDCD, and BCBC is not parallel to ADAD. Variable points P,Q,R,SP, Q, R, S are taken on AB,BC,CD,DAAB, BC, CD, DA respectively so that PQRSPQRS is a parallelogram. Find the locus of its center.
geometryparallelogramgeometry unsolved
$1993$ points are arranged in a circle

Source: 31-th Vietnamese Mathematical Olympiad 1993

2/17/2007
19931993 points are arranged in a circle. At time 00 each point is arbitrarily labeled +1+1 or −1-1. At times n=1,2,3,...n = 1, 2, 3, ... the vertices are relabeled. At time nn a vertex is given the label +1+1 if its two neighbours had the same label at time n−1n-1, and it is given the label −1-1 if its two neighbours had different labels at time n−1n-1. Show that for some time n>1n > 1 the labeling will be the same as at time 1.1.
combinatorics unsolvedcombinatorics