2017 AMC 10B Problems/Problem 3
Problem
Real numbers , , and satify the inequalities , , and . Which of the following numbers is necessarily positive?
Solution
We start from the last answer choice because the answer for these type of questions are usually in the last few answer choices. Notice that , the last answer choice, 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.
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.