Difference between revisions of "2010 AMC 12B Problems/Problem 23"
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Revision as of 10:00, 4 July 2013
Problem 23
Monic quadratic polynomial 
and 
have the property that 
has zeros at 
and 
, and 
has zeros at 
and 
. What is the sum of the minimum values of and ?
Solution
. Notice that has roots 
, so that the roots of are the roots of 
. For each individual equation, the sum of the roots will be 
(symmetry or Vieta's). Thus, we have 
, or 
. Doing something similar for gives us 
.
We now have 
. Since 
is monic, the roots of 
are "farther" from the axis of symmetry than the roots of 
. Thus, we have 
, or 
. Adding these gives us 
, or 
. Plugging this into 
, we get 
.
The minimum value of is 
, and the minimum value of is 
. Thus, our answer is 
, or answer 
.
See Also
| 2010 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 22 |
Followed by Problem 24 |
| 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. 
