Name:      

Directions: Show all work.

1. [4 points] Let $S$ be a set of $5$ points in an equilateral triangle with side length $2$. Prove that $S$ contains a pair of points that are at distance at most $1$.

   

2. [2 parts, 3 points each] Recall that two integers $x$ and $y$ are relatively prime if their greatest common divisor is $1$.

   1. Find a maximum size subset of $\{1,\dots ,8\}$ such that no two elements in $A$ are relatively prime. Prove your answer is correct.
   2. Find a maximum size subset of $\{1,\dots ,2n\}$ such that no two elements in $A$ are relatively prime. Prove your answer is correct.