Difference between revisions of "2015 AIME II Problems/Problem 14"
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+ | The expression we want to find is <math>2(x^3+y^3) + x^3y^3</math>. | ||
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+ | Factor the given equations as <math>x^4y^4(x+y) = 810</math> and <math>x^3y^3(x^3+y^3)=945</math>, respectively. Dividing the latter by the former equation yields <math>\frac{x^2-xy+y^2}{xy} = \frac{945}{810}</math>. Adding 3 to both sides and simplifying yields <math>\frac{(x+y)^2}{xy} = \frac{25}{6}</math>. Solving for <math>x+y</math> and substituting this expression into the first equation yields <math>\frac{5\sqrt{6}}{6}(xy)^{\frac{9}{2}} = 810</math>. Solving for <math>xy</math>, we find that <math>xy = 3\sqrt[3]{2}</math>, so <math>x^3y^3 = 54</math>. Substituting this into the second equation and solving for <math>x^3+y^3</math> yields <math>x^3+y^3=\frac{35}{2}</math>. So, the expression to evaluate is equal to <math>2 \times \frac{35}{2} + 54 = \boxed{89}</math>. |
Revision as of 21:52, 26 March 2015
Problem
Let and
be real numbers satisfying
and
. Evaluate
.
Solution
The expression we want to find is .
Factor the given equations as and
, respectively. Dividing the latter by the former equation yields
. Adding 3 to both sides and simplifying yields
. Solving for
and substituting this expression into the first equation yields
. Solving for
, we find that
, so
. Substituting this into the second equation and solving for
yields
. So, the expression to evaluate is equal to
.