4. On a plane, there are 10 different points. We considered the midpoints of all segments connecting all pairs of points. What is the smallest number of midpoints that could have arisen?
4. Answer: 17 midpoints.

Consider the case when the points lie on a single line. Introduce coordinates on it, and let the points be $a_{1}<\cdots<a_{10}$. Consider the midpoints of segments involving $a_{1}$. There are nine of them, and they form an increasing sequence $\frac{a_{1}+a_{2}}{2}<\frac{a_{1}+a_{3}}{2}<\cdots<\frac{a_{1}+a_{10}}{2}$. Similarly, consider the midpoints of segments involving $a_{10}$. There are also nine of them, and they form a sequence $\frac{a_{1}+a_{10}}{2}<\frac{a_{2}+a_{10}}{2}<\cdots<\frac{a_{9}+a_{10}}{2}$. The largest number of the first sequence coincides with the smallest number of the second, and together we have 17 pairwise distinct midpoints. Thus, there are at least 17 midpoints in total.

If we take points with even coordinates $2,4,6, \ldots, 20$, the midpoints will be all points with integer coordinates from 3 to 19 inclusive, and there are exactly 17 of them.

Now consider the general case of the points' arrangement on the plane. Draw lines through all pairs of points, and consider a line that is not perpendicular to any of these lines. Then, in the projection onto this line, all points will be distinct. The midpoint under projection becomes a midpoint, and for points on a line, we have at least 17 distinct midpoints. Therefore, there were at least 17 midpoints in total.