Twelve straight lines are drawn in a plane such that no two of them are parallel and no three of them are concurrent. A circle is now drawn in the same plane such that all the points of intersection of all the lines lie inside the circle. What is the number of non-overlapping regions into which the circle is divided

The problem can be solved by using the principle of mathematical induction.

1. Start with a circle and no lines. There is only 1 region.

2. Add the first line. It divides the circle into 2 regions.

3. Add the second line. It intersects the first line once, and each intersection divides a region into two. So, we get 2 + 1 = 3 regions.

4. Add the third line. It intersects the first two lines twice, and each intersection divides a region into two. So, we get 3 + 2 = 5 regions.

5. Add the fourth line. It intersects the first three lines three times, and each intersection divides a region into two. So, we get 5 + 3 = 8 regions.

6. Continue this pattern. Each new line intersects all the previous lines once more than the last line did, and each intersection divides a region into two.

So, the number of regions for n lines is given by the formula:

1 + (1 + 2 + 3 + ... + n) = 1 + n*(n+1)/2

For 12 lines, the number of regions is:

1 + 12*(12+1)/2 = 1 + 78 = 79

So, the circle is divided into 79 non-overlapping regions.