MathDB

5

Part of 2019 Czech-Polish-Slovak Junior Match

Problems(2)

everyone has d friends, every 2 strangers have exactly one common friend

Source: Czech-Polish-Slovak Match Junior 2019, individual p5 CPSJ

1/20/2020
Given is a group in which everyone has exactly dd friends and every two strangers have exactly one common friend. Prove that there are at most d2+1d^2 + 1 people in this group.
combinatorics
120 rotations around it's center of a regular 360gon

Source: Czech-Polish-Slovak Match Junior 2019, team p5 CPSJ

1/20/2020
Let A1A2...A360A_1A_2 ...A_{360} be a regular 360360-gon with centre SS. For each of the triangles A1A50A68A_1A_{50}A_{68} and A1A50A69A_1A_{50}A_{69} determine, whether its images under some 120120 rotations with centre SS can have (as triangles) all the 360360 points A1,A2,...,A360A_1, A_2, ..., A_{360} as vertices.
geometrygeometric transformationrotationcombinatorial geometrycombinatorics