Difference between revisions of "2021 Fall AMC 12A Problems/Problem 19"
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+ | |||
+ | == Solution 2 == | ||
+ | For choice A, we have | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | \left| \sin x = \sin \left( x^2 \right) \right| | ||
+ | & = \left| \sin 10^\circ = \sin \left( \left( 10^2 \right)^\circ \right) \right| \\ | ||
+ | & = \left| \sin 10^\circ - \sin 100^\circ \right| \\ | ||
+ | & = \left| \sin 10^\circ - \sin 80^\circ \right| \\ | ||
+ | & = \sin 80^\circ - \sin 10^\circ . | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | For choice B, we have | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | \left| \sin x = \sin \left( x^2 \right) \right| | ||
+ | & = \left| \sin 13^\circ = \sin \left( \left( 13^2 \right)^\circ \right) \right| \\ | ||
+ | & = \left| \sin 13^\circ - \sin 169^\circ \right| \\ | ||
+ | & = \left| \sin 10^\circ - \sin 11^\circ \right| \\ | ||
+ | & = \sin 11^\circ - \sin 10^\circ . | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | For choice C, we have | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | \left| \sin x = \sin \left( x^2 \right) \right| | ||
+ | & = \left| \sin 14^\circ = \sin \left( \left( 14^2 \right)^\circ \right) \right| \\ | ||
+ | & = \left| \sin 14^\circ - \sin 196^\circ \right| \\ | ||
+ | & = \left| \sin 14^\circ + \sin 16^\circ \right| \\ | ||
+ | & = \sin 14^\circ + \sin 16^\circ . | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | For choice D, we have | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | \left| \sin x = \sin \left( x^2 \right) \right| | ||
+ | & = \left| \sin 19^\circ = \sin \left( \left( 19^2 \right)^\circ \right) \right| \\ | ||
+ | & = \left| \sin 19^\circ - \sin 361^\circ \right| \\ | ||
+ | & = \left| \sin 19^\circ - \sin 1^\circ \right| \\ | ||
+ | & = \sin 19^\circ - \sin 1^\circ . | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | For choice E, we have | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | \left| \sin x = \sin \left( x^2 \right) \right| | ||
+ | & = \left| \sin 20^\circ = \sin \left( \left( 20^2 \right)^\circ \right) \right| \\ | ||
+ | & = \left| \sin 20^\circ - \sin 400^\circ \right| \\ | ||
+ | & = \left| \sin 20^\circ - \sin 40^\circ \right| \\ | ||
+ | & = \sin 40^\circ - \sin 20^\circ . | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | Therefore, the answer is <math>\boxed{\textbf{(B) }13}</math>. | ||
+ | |||
+ | ~Steven Chen (www.professorchenedu.com) | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2021 Fall|ab=A|num-b=18|num-a=20}} | {{AMC12 box|year=2021 Fall|ab=A|num-b=18|num-a=20}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 20:51, 25 November 2021
Contents
Problem 19
Let be the least real number greater than such that , where the arguments are in degrees. What is rounded up to the closest integer?
Solution 1
The smallest to make would require , but since needs to be greater than , these solutions are not valid.
The next smallest would require , or .
After a bit of guessing and checking, we find that , and , so the solution lies between and , making our answer
Note: One can also solve the quadratic and estimate the radical.
~kingofpineapplz
Solution 2
For choice A, we have
For choice B, we have
For choice C, we have
For choice D, we have
For choice E, we have
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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