Kalinin D.A.

On a line, several points were marked. After that, between each pair of adjacent points, one more point was marked. This "densification" was repeated two more times (a total of three times). As a result, 113 points were marked on the line. How many points were initially marked?
Find how many points there were before the last compaction, i.e., solve the problem "from the end".

## Solution

If there were $n$ points marked (before compaction), then after compaction, there will be $2 n-1$ points (of which $n$ are old and $n-1$ are new). If after compaction, there are $k$ points, then $2 n-1=k$ or $n=(k+1) / 2$. Therefore, before the last compaction, there were $(113+1) / 2=57$ points, before the second $(57+1) / 2=29$ points, and at the very beginning $(29+1) / 2=15$ points.

## Answer

15 points.