1985 AMC 12 第 10 题

Question

任意一个圆与函数

y=\sin x

的图像的交点数可以是

An arbitrary circle can intersect the graph of

y=\sin x

in

至多

2

个点

at most

2

points

至多

4

个点

at most

4

points

至多

6

个点

at most

6

points

至多

8

个点

at most

8

points

超过

16

个点

more than

16

points

答案：E

知识点：圆交点计数三角学

难度评级：2240

小提示：

考虑一个在原点与

x

轴相切的圆

Consider a circle tangent to the

x

-axis at the origin

大提示：

半径非常大时，在许多个正弦波范围内，圆的下弧都靠近坐标轴

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

解答：

取一个在原点与

x

轴相切、圆心位于坐标轴上方很高处的圆。随着半径增大，在任意长的区间内，它的下弧都保持为正且可以任意接近

x

轴。在

y=\sin x

的每个正弧上，正弦曲线在两端为

0

，而在中间远高于这段圆弧，从而产生两个交点。因此，选取足够大的半径便可得到超过

16

个交点。

所以正确答案是 E。

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 第 10 题

答案：E

难度评级：2240

小提示：

大提示：

解答：

其他年份的第 10 题