Olympiad problems

ICMC 4, 1

A set of points in the plane is called [i]sane[/i] 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 $\mathcal{P}$? (b) Does there exist an uncountably infinite sane set $\mathcal{Q}$? [i]Proposed by Tony Wang[/i]

2020 LIMIT Category 2, 20

Tags: countable , convergence , sequence , calculus

Let $\{a_n \}_n$ be a sequence of real numbers such there there are countably infinite distinct subsequences converging to the same point. We call two subsequences distinct if they do not have a common term. Which of the following statements always holds: (A) $\{a_n \}_n$ is bounded (B) $\{a_n \}_n$ is unbounded (C) The set of convergent subsequence $\{a_n \}_n$ is countable (D) None of these

1998 IMC, 5

Tags: countable

$S$ is a family of balls in $\mathbb{R}^{n}$ ($n > 1$) such that the intersection of any two contains at most one point. Show that the set of points belonging to at least two members of $S$ is countable.