Difference between revisions of "2019 AMC 12A Problems/Problem 17"
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==Solution 1== | ==Solution 1== | ||
− | Applying | + | Applying [https://artofproblemsolving.com/wiki/index.php/Newton's_Sums Newton's Sums], we have<cmath>s_{k+1}+(-5)s_k+(8)s_{k-1}+(-13)s_{k-2}=0,</cmath>so<cmath>s_{k+1}=5s_k-8s_{k-1}+13s_{k-2},</cmath>we get the answer as <math>5+(-8)+13=10</math>. |
==Solution 2== | ==Solution 2== |
Revision as of 21:18, 12 March 2023
Contents
[hide]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, 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
Solution 4
Let be the roots of
. Then:
\
\
If we multiply both sides of the equation by , where
is a positive integer, then that won't change the coefficients, but just the degree of the new polynomial and the other term's exponents. We can try multiplying to find
, but that is just to check. So then with the above information about
, we see that:
,
,
Then:
This means that , as expected. So we have
. So our answer is
-IzhanAli
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.