2010 AMC 12B Problems/Problem 23
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.