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 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
is incorrect; picking
reveals that
is 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 |
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