Question 22
Daniel chooses a positive integer $n$ and tells it to Ana. With this information, Ana chooses a positive integer $k$ and tells it to Daniel. Daniel draws $n$ circles on a piece of paper and chooses $k$ different points with the condition that each of them belongs to one of the circles he drew. Then he erases the circles, and only the $k$ points he marked remain visible. From these points, Ana must reconstruct at least one of the circles that Daniel drew. Determine what is the smallest value of $k$ that allows Ana to achieve her goal regardless of how Daniel chooses the $n$ circumferences and the $k$ points.