Difference between revisions of "2014 AIME II Problems/Problem 5"
Line 4: | Line 4: | ||
==Solution== | ==Solution== | ||
− | Let r, s, -r-s be the roots of p(x) (per Vieta's). Then <math>r^3 + ar + b = 0</math> and similarly for s. Also, <cmath>q(r+4) = (r+4)^3 + a(r+4) + b + 240 = 12r^2 + 48r + 304 + 4a = 0</cmath> Set up a similar equation for s: <cmath>q(s-3) = (s-3)^3 + a(s-3) + b + 240 = -9s^2 + 27s + 213 - 3a = 0</cmath>. Simplifying and adding the equations gives <cmath>3r^2 - 3s^2 + 12r + 9s + 147 = 0 (*)</cmath> Now, let's deal with the a*x. Equating the a in both equations (per Vieta) <cmath>rs + (-r-s)(r+s) = (r+4)(s-3) + (-r-s-1)(r+s+1)</cmath> | + | Let r, s, -r-s be the roots of p(x) (per Vieta's). Then <math>r^3 + ar + b = 0</math> and similarly for s. Also, |
+ | <cmath>q(r+4) = (r+4)^3 + a(r+4) + b + 240 = 12r^2 + 48r + 304 + 4a = 0</cmath> | ||
+ | |||
+ | Set up a similar equation for s: | ||
+ | |||
+ | <cmath>q(s-3) = (s-3)^3 + a(s-3) + b + 240 = -9s^2 + 27s + 213 - 3a = 0</cmath>. | ||
+ | |||
+ | Simplifying and adding the equations gives | ||
+ | <cmath>3r^2 - 3s^2 + 12r + 9s + 147 = 0</cmath> | ||
+ | |||
+ | <cmath>r^2 - s^2 + 4r + 3s + 49 = 0 (*)</cmath> | ||
+ | |||
+ | Now, let's deal with the a*x. Equating the a in both equations (per Vieta) | ||
+ | <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 <math>r = -5</math> and <math>r = 1</math>, corresponding to <math>s = -6</math> and <math>s = 9</math>, and <math>|b| = 330, 90</math>, for an answer of <math>\boxed{420}</math>. |
Revision as of 11:45, 31 March 2014
Problem 5
Real numbers and are roots of , and and are roots of . Find the sum of all possible values of .
Solution
Let r, s, -r-s be the roots of p(x) (per Vieta's). Then and similarly for s. Also,
Set up a similar equation for s:
.
Simplifying and adding the equations gives
Now, let's deal with the a*x. Equating the a in both equations (per Vieta) which eventually simplifies to
Substitution into (*) should give and , corresponding to and , and , for an answer of .