2014 AIME II Problems/Problem 5
Problem 5
Real numbers and
are roots of
, and
and
are roots of
. Find the sum of all possible values of
.
Solution
Let ,
, and
be the roots of
(per Vieta's). Then
and similarly for
. Also,
Set up a similar equation for :
Simplifying and adding the equations gives
Now, let's deal with the terms. Plugging the roots
,
, and
into
yields a long polynomial, and plugging the roots
,
, and
into $q(x) yields another long polynomial. Equating the coefficients of x in both polynomials:
<cmath>rs + (-r-s)(r+s) = (r+4)(s-3) + (-r-s-1)(r+s+1),</cmath>
which eventually simplifies to
<cmath>s = \frac{13 + 5r}{2}.</cmath>
Substitution into (*) should give$ (Error compiling LaTeX. Unknown error_msg)r = -5r = 1
s = -6
s = 9
|b| = 330, 90
\boxed{420}$.
See also
2014 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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