MathDB

1

Part of ICMC 4

Problems(2)

Countably many sane points

Source: ICMC 2021 Round 1 P1

3/1/2022
A set of points in the plane is called sane if no three points are collinear and the angle between any three distinct points is a rational number of degrees.
(a) Does there exist a countably infinite sane set P\mathcal{P}?
(b) Does there exist an uncountably infinite sane set Q\mathcal{Q}?
Proposed by Tony Wang
geometryCountableuncountableICMCcollege contests
union closed subsets

Source: ICMC 2021 Round 2 P1

12/12/2021
Let SS be a set with 10 distinct elements. A set TT of subsets of SS (possibly containing the empty set) is called union-closed if, for all A,B∈TA, B \in T, it is true that A∪B∈TA \cup B \in T. Show that the number of union-closed sets TT is less than 210232^{1023}.
Proposed by Tony Wang
ICMCcollege contestsSetscombinatoricscounting