AMC10 2009 A

AMC10 2009 A · Q19

AMC10 2009 A · Q19. It mainly tests Circle theorems, Divisibility & factors.

Circle $A$ has radius 100. Circle $B$ has an integer radius $r < 100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many possible values can $r$ have?

圆 $A$ 的半径为100。圆 $B$ 的整数半径 $r < 100$，它在绕圆 $A$ 的周长滚一圈时始终与圆 $A$ 内部相切。在 $B$ 滚完一圈的开始和结束时，两圆有相同的切点。$r$ 有多少个可能值？

(A) 4 4

(B) 8 8

(D) 50 50

(E) 90 90

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Circles $A$ and $B$ have circumferences $200\pi$ and $2\pi r$, respectively. After circle $B$ begins to roll, its initial point of tangency with circle $A$ touches circle $A$ again a total of \[ \frac{200\pi}{2\pi r}=\frac{100}{r} \] times. In order for this to be an integer greater than 1, $r$ must be one of the integers 1, 2, 4, 5, 10, 20, 25, or 50. Hence there are a total of 8 possible values of $r$.

答案（B）：圆 $A$ 和圆 $B$ 的周长分别是 $200\pi$ 和 $2\pi r$。当圆 $B$ 开始滚动后，它与圆 $A$ 的初始切点再次接触圆 $A$ 的总次数为 \[ \frac{200\pi}{2\pi r}=\frac{100}{r} \] 次。为了使该次数为大于 1 的整数，$r$ 必须是整数 1、2、4、5、10、20、25 或 50。因此，$r$ 一共有 8 个可能的取值。

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