2021 Fall AMC 12A Problems/Problem 19

Problem 19

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?

$\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 14 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 20$

Solution 1

The smallest $x$ to make $\sin(x) = \sin(x^2)$ would require $x=x^2$ , but since $x$ needs to be greater than $1$ , these solutions are not valid.

The next smallest $x$ would require $x=180-x^2$ , or $x^2+x=180$ .

After a bit of guessing and checking, we find that $12^2+12=156$ , and $13^2+13=182$ , so the solution lies between $12{ }$ and $13$ , making our answer $\boxed{\textbf{(B) } 13}.$

Note: One can also solve the quadratic and estimate the radical.

~kingofpineapplz

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2021 Fall AMC 12A Problems/Problem 19

Problem 19

Solution 1