2014 AMC 10A Problems/Problem 7
Problem
Nonzero real numbers ,
,
, and
satisfy
and
. How many of the following inequalities must be true?
Solution
First, we note that must be true by adding our two original inequalities.
Though one may be inclined to think that must also be true, it is not, for we cannot subtract inequalities.
In order to prove that the other inequalities are false, we only need to provide one counterexample. Let's try substituting that gives
, which is not the case, thus
is false.
states that
. This is also false, thus
is false.
states that
. This is false, so
is false.
One of our four inequalities is true, hence, our answer is
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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 |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.