Difference between revisions of "2019 AMC 12A Problems/Problem 17"
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So, <math>a+ b + c = 5 -8 + 13 = \boxed{\textbf{(D) } 10}</math>. | So, <math>a+ b + c = 5 -8 + 13 = \boxed{\textbf{(D) } 10}</math>. | ||
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+ | -gregwwl | ||
==Video Solution== | ==Video Solution== |
Revision as of 20:22, 15 March 2019
Problem
Let denote the sum of the
th powers of the roots of the polynomial
. In particular,
,
, and
. Let
,
, and
be real numbers such that
for
,
,
What is
?
Solution 1
Applying Newton's Sums (see this link), we haveso
we get the answer as
.
Solution 2
Let , and
be the roots of the polynomial. Then,
Adding these three equations, we get
can be written as
, giving
We are given that is satisfied for
,
,
, meaning it must be satisfied when
, giving us
.
Therefore, , and
by matching coefficients.
.
Solution 3
Let , and
be the roots of the polynomial. By Vieta's Formulae, we have
.
We know . Consider
.
Using and
, we see
.
We have
Rearrange to get
So, .
-gregwwl
Video Solution
For those who want a video solution: https://www.youtube.com/watch?v=tAS_DbKmtzI
See Also
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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.