Difference between revisions of "2016 AMC 10B Problems/Problem 3"
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<math>\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048</math> | <math>\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048</math> | ||
− | ==Solution== | + | ==Solution 1== |
Substituting carefully, <math>\Bigg\vert\Big\vert 2016-(-2016)\Big\vert-2016\Bigg\vert-(-2016)</math> | Substituting carefully, <math>\Bigg\vert\Big\vert 2016-(-2016)\Big\vert-2016\Bigg\vert-(-2016)</math> | ||
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Substitute <math>-y = x = -2016</math> into the equation. Now, it is <math>\Bigg\vert\Big\vert |y|+y\Big\vert-|y|\Bigg\vert+y</math>. Since <math>y = 2016</math>, it is a positive number, so <math>|y| = y</math>. Now the equation is <math>\Bigg\vert\Big\vert y+y\Big\vert-y\Bigg\vert+y</math>. This further simplifies to <math>2y-y+y = 2y</math>, so the answer is <math>\boxed{\textbf{(D)}\ 4032}</math> | Substitute <math>-y = x = -2016</math> into the equation. Now, it is <math>\Bigg\vert\Big\vert |y|+y\Big\vert-|y|\Bigg\vert+y</math>. Since <math>y = 2016</math>, it is a positive number, so <math>|y| = y</math>. Now the equation is <math>\Bigg\vert\Big\vert y+y\Big\vert-y\Bigg\vert+y</math>. This further simplifies to <math>2y-y+y = 2y</math>, so the answer is <math>\boxed{\textbf{(D)}\ 4032}</math> | ||
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+ | ==Video Solution (CREATIVE THINKING)== | ||
+ | https://youtu.be/2cPRcI_rIvU | ||
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+ | ~Education, the Study of Everything | ||
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==Video Solution== | ==Video Solution== |
Latest revision as of 11:48, 2 July 2023
Contents
Problem
Let . What is the value of ?
Solution 1
Substituting carefully,
becomes which is .
Solution 2
Solution by e_power_pi_times_i
Substitute into the equation. Now, it is . Since , it is a positive number, so . Now the equation is . This further simplifies to , so the answer is
Video Solution (CREATIVE THINKING)
~Education, the Study of Everything
Video Solution
~savannahsolver
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.