Difference between revisions of "2019 AMC 12A Problems/Problem 17"
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==Solution 1== | ==Solution 1== | ||
− | Applying Newton's Sums (see [https://artofproblemsolving.com/wiki/index.php/Newton%27s_Sums this link]), we get the answer as <math>10</math>. | + | Applying Newton's Sums (see [https://artofproblemsolving.com/wiki/index.php/Newton%27s_Sums this link]), 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 00:24, 19 February 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 havesowe 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, .
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 |
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