Lines and Circles

Worked solution

The answer is 80. Count the maximum for each kind of pair. Two lines meet in at most 1 point: $\binom{5}{2} = 10$. Two circles meet in at most 2: $2\binom{5}{2} = 20$. A line and a circle meet in at most 2: $2 \times 5 \times 5 = 50$. That bounds the total by $10 + 20 + 50 = 80$. The bound is achieved: take five circles of nearly equal radius with centres close together (so every two cross twice) and five lines through the common central region in general directions (so each crosses every circle twice and no two are parallel). A small perturbation makes all 80 intersection points distinct.