Olympiad problems

2020 USA TSTST, 3

Tags: niven theorem

We say a nondegenerate triangle whose angles have measures $\theta_1$, $\theta_2$, $\theta_3$ is [i]quirky[/i] if there exists integers $r_1,r_2,r_3$, not all zero, such that \[r_1\theta_1+r_2\theta_2+r_3\theta_3=0.\] Find all integers $n\ge 3$ for which a triangle with side lengths $n-1,n,n+1$ is quirky. [i]Evan Chen and Danielle Wang[/i]

2013 USA TSTST, 9

Tags: rotation , invariant , niven theorem , geocombontturnedintoalg , trigonometry , geometry , inequalities

Let $r$ be a rational number in the interval $[-1,1]$ and let $\theta = \cos^{-1} r$. Call a subset $S$ of the plane [i]good[/i] if $S$ is unchanged upon rotation by $\theta$ around any point of $S$ (in both clockwise and counterclockwise directions). Determine all values of $r$ satisfying the following property: The midpoint of any two points in a good set also lies in the set.

2020 India National Olympiad, 5

Tags: tiling , descent , niven theorem , rational , galois theory , trigonometry , number theory

Infinitely many equidistant parallel lines are drawn in the plane. A positive integer $n \geqslant 3$ is called frameable if it is possible to draw a regular polygon with $n$ sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon. (a) Show that $3, 4, 6$ are frameable. (b) Show that any integer $n \geqslant 7$ is not frameable. (c) Determine whether $5$ is frameable. [i]Proposed by Muralidharan[/i]