1971 AMC 12 Problem 17

Question

A circular disk is divided by

2n

equally spaced radii

(n\gt0)

and one secant line. The maximum number of nonoverlapping areas into which the disk can be divided is:

2n+1

2n+2

3n-1

3n

3n+1

Begin with the

2n

sectors made by the radii

A secant can cross at most

n

of the radii, so count the pieces of the secant chord

The radii first make

2n

sectors. A secant not through the center can meet at most one radius in each of

n

opposite pairs, hence at most

n

radii. Those intersections divide the secant chord into

n+1

pieces, each of which adds one region. The maximum is therefore

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

Therefore, the correct answer is E.

Problem 17 in other years