2021 Fall AMC 12A Problem 19

Problem 19 of 25HarderAlgebra Geometry

Let

x

be the least real number greater than

1

such that

\sin(x) = \sin(x^2),

where the arguments are in degrees. What is

x

rounded up to the closest integer?

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Solution

Equal sines require

x^2 = x + 360k

or

x^2 = 180 - x + 360k

for some integer

k.

The family

x^2 = x + 360k

first exceeds

1

at

k = 1,

giving

x \approx 19.5.

The family

x^2 = 180 - x + 360k

with

k = 0

gives

x^2 + x - 180 = 0,

so

x = \dfrac{-1 + \sqrt{721}}{2} \approx 12.93,

which is smaller. Rounded up,

x = 13.

Thus, the correct answer is B.

Tagged: trigonometry · quadratic

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