1985 AHSME Problems/Problem 10

Problem

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

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

Solution

Consider a circle whose center lies on the positive $y$ -axis and which passes through the origin. As the radius of this circle becomes arbitrarily large, its curvature near the $x$ -axis becomes almost flat, and so it can intersect the curve $y = \sin x$ arbitrarily many times (since the $x$ -axis itself intersects the curve infinitely many times). Hence, in particular, we can choose a radius sufficiently large that the circle intersects the curve at $\boxed{\text{(E)} \ \text{more than }16 \text{ points}}$ .

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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