1985 AHSME Problems/Problem 10

Problem

An arbitrary circle can intersect the graph of $y=\sin x$ in

$\mathrm{(A) \ } \text{at most }2\text{ points} \qquad \mathrm{(B) \ }\text{at most }4\text{ points} \qquad \mathrm{(C) \ } \text{at most }6\text{ points} \qquad \mathrm{(D) \ } \text{at most }8\text{ points}\qquad \mathrm{(E) \ }\text{more than }16\text{ points}$

Solution

Consider a circle with center on the positive y-axis and that passes through the origin. As the radius of this circle becomes arbitrarily large, its shape near the x-axis becomes very similar to that of $y=0$ , which intersects $y=\sin x$ infinitely many times. It therefore becomes obvious that we can pick a radius large enough so that the circle intersects the sinusoid as many times as we wish, $\boxed{\text{E}}$ .

1985 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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1985 AHSME Problems/Problem 10

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