2000 AMC 10 Problems/Problem 24
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Problem
Let 
be a function for which 
. Find the sum of all values of 
for which 
.
Solution
In the definition of , let 
. We get: 
. As we have , we must have 
, in other words 
.
One can now either explicitly compute the roots, or use Vieta's formulas. According to them, the sum of the roots of 
is 
. In our case this is 
.
(Note that for the above approach to be completely correct, we should additionally verify that there actually are two distinct real roots. This is, for example, obvious from the facts that 
and 
.)
See Also
| 2000 AMC 10 (Problems • Resources) | ||
| Preceded by Problem 23 | Followed by Problem 25 | |
| 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 | ||
