MathDB

3

Part of 2013 Czech-Polish-Slovak Junior Match

Problems(2)

AK = KL inside a cyclic pentagon ABCDE with AB = BC = CD

Source: Czech-Polish-Slovak Junior Match 2013, Individual p3 CPSJ

3/14/2020
The ABCDEABCDE pentagon is inscribed in a circle and AB=BC=CDAB = BC = CD. Segments ACAC and BEBE intersect at KK, and Segments ADAD and CECE intersect at pointL L. Prove that AK=KLAK = KL.
equal segmentsgeometrypentagonCyclic
5 of n people may sit at a circle on a table among friends / strangers

Source: Czech-Polish-Slovak Junior Match 2013, Team p3 CPSJ

3/21/2020
In a certain group there are n≥5n \ge 5 people, with every two people who do not know each other exactly having one mutual friend and no one knows everyone else. Prove 55 of nn people, may sit at a circle around the table so that each of them sits between a) friends, b) strangers.
combinatorics