Difference between revisions of "2021 AMC 12A Problems/Problem 2"
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==Solution 3 (Graphing)== | ==Solution 3 (Graphing)== | ||
If we graph <math>\sqrt{x^2+y^2}=x+y,</math> then we get the positive <math>x</math>-axis and the positive <math>y</math>-axis, plus the origin. Therefore, the answer is <math>\boxed{\textbf{(D)}}.</math> | If we graph <math>\sqrt{x^2+y^2}=x+y,</math> then we get the positive <math>x</math>-axis and the positive <math>y</math>-axis, plus the origin. Therefore, the answer is <math>\boxed{\textbf{(D)}}.</math> | ||
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+ | Graph in Desmos: https://www.desmos.com/calculator/rlz6actjrw | ||
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~MRENTHUSIASM (credit given to TheAMCHub) | ~MRENTHUSIASM (credit given to TheAMCHub) | ||
Revision as of 16:14, 12 February 2021
Contents
Problem
Under what conditions does hold, where
and
are real numbers?
It is never true.
It is true if and only if
.
It is true if and only if
.
It is true if and only if
and
.
It is always true.
Solution 1
Square both sides to get . Then,
. Also, it is clear that both sides of the equation must be nonnegative. The answer is
.
Solution 2 (Quick Inspection)
The left side of the original equation is the arithmetic square root, which is always nonnegative. So, we need which eliminates
and
Next, picking
reveals that
and
are both incorrect. By POE (Process of Elimination), the answer is
~MRENTHUSIASM
Solution 3 (Graphing)
If we graph then we get the positive
-axis and the positive
-axis, plus the origin. Therefore, the answer is
Graph in Desmos: https://www.desmos.com/calculator/rlz6actjrw
~MRENTHUSIASM (credit given to TheAMCHub)
Video Solution by Hawk Math
https://www.youtube.com/watch?v=P5al76DxyHY
Video Solution by OmegaLearn (Using logic and analyzing answer choices)
~ pi_is_3.14
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 1 |
Followed by Problem 3 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.