Let $a$ denote the smallest number of circles of radius 1 that can completely cover a given polygon $M$, and $b$ - the largest number of non-overlapping circles of radius 1 with centers inside the polygon $M$. Which number is greater, $a$ or $b$?
Answer: $a \geq b$. Consider an arbitrary arrangement of non-intersecting circles of radius 1 with centers (denoted as $A_{\mathrm{i}}$), lying inside the polygon $M$. Then all $A_{\mathrm{i}}$ lie inside the polygon $M$ and the distance between any two of them is greater than 2. This means that in any covering of the polygon $M$ by circles of radius 1, no more than one of the points $A_{\mathrm{i}}$ can be contained in any single circle. Thus, we have obtained that $a \geq b$.