*Challenge question #28 (12/09/2016)*

**Junior Question**

A regular hexagon and an equilateral triangle have the same perimeter.

What is the ratio of their areas?

**Senior Question**

Let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$.

Consider the values $\lfloor 2x \rfloor$, $2 \lfloor x \rfloor$,
and $\lfloor x+\frac{1}{2} \rfloor + \lfloor x-\frac{1}{2} \rfloor$.

Show that, for any value of $x$, two of these values will be the same, and the third will be different.