2003 AMC 12B Problem 23

Problem 23 of 25HarderAlgebra Geometry Counting & Probability

The number of

x

-intercepts on the graph of

y = \sin(\frac{1}{x})

in the interval

(0.0001, 0.001)

is closest to

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Solution

The intercepts occur where

\frac{1}{x} = k\pi,

that is

x = \dfrac{1}{k\pi}

for a nonzero integer

k.

The condition

0.0001 \lt \dfrac{1}{k\pi} \lt 0.001

becomes

\frac{1000}{\pi} \lt k \lt \frac{10000}{\pi}.

The number of such integers is

\begin{aligned} &\left\lfloor \frac{10000}{\pi} \right\rfloor - \left\lfloor \frac{1000}{\pi} \right\rfloor \\ &= 3183 - 318 = 2865, \end{aligned}

closest to

2900.

Thus, the correct answer is A.

More practice

Concepts: counting integers in a range · floor and ceiling functions · trigonometry

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.