1985 AMC 12 Problem 10

Question

An arbitrary circle can intersect the graph of

y=\sin x

in

at most

2

points

at most

4

points

at most

6

points

at most

8

points

more than

16

points

Answer: E

Concepts:circle counting intersections trigonometry

Difficulty rating: 2240

Small hint:

Consider a circle tangent to the

x

-axis at the origin

Big hint:

A very large radius keeps the lower arc close to the axis across many sine waves

Solution:

Take a circle tangent to the

x

-axis at the origin with its center high above the axis. As its radius grows, its lower arc stays positive but arbitrarily close to the

x

-axis over an arbitrarily long interval. On each positive arch of

y=\sin x,

the sine curve is

0

at the endpoints and rises well above this circular arc in between, producing two crossings. Choosing a sufficiently large radius therefore produces more than

16

intersections.

Thus the correct answer is E.

1985 AMC 12 Problem 10

Answer: E

Difficulty rating: 2240

Small hint:

Big hint:

Solution:

Problem 10 in other years