2021 AMC 12A Problems/Problem 2
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
Square both sides to get . Then,
. Then, the answer is
. Consider a right triangle with legs
and
and hypotenuse
. Then one of the legs must be equal to
, but they are also nonnegative as they are lengths. Therefore, both
and
are correct.
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
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 |
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