Difference between revisions of "2017 AMC 10B Problems/Problem 3"
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~savannahsolver | ~savannahsolver | ||
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+ | ==Video Solution by TheBeautyofMath== | ||
+ | https://youtu.be/zTGuz6EoBWY?t=525 | ||
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+ | ~IceMatrix | ||
==See Also== | ==See Also== |
Latest revision as of 03:25, 4 December 2020
Problem
Real numbers , , and satisfy the inequalities , , and . Which of the following numbers is necessarily positive?
Solution
Notice that must be positive because . Therefore the answer is .
The other choices:
As grows closer to , decreases and thus becomes less than .
can be as small as possible (), so grows close to as approaches .
For all , , and thus it is always negative.
The same logic as above, but when this time.
Video Solution
~savannahsolver
Video Solution by TheBeautyofMath
https://youtu.be/zTGuz6EoBWY?t=525
~IceMatrix
See Also
2017 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 |
2017 AMC 12B (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.