Difference between revisions of "2015 AIME II Problems/Problem 14"
(→Solution 3) |
m (→Solution 3) |
||
Line 13: | Line 13: | ||
==Solution 3== | ==Solution 3== | ||
− | Add three times the first equation to the second equation and factor to get <math>(xy)^3(x^3+3x^2y+3xy^2+y^3)=(xy)^3(x+y)^3=3375</math>. Taking the cube root yields <math>xy(x+y)=15</math>. Noting that the first equation is <math>(xy)^3\cdot(xy(x+y))=810</math>, we find that <math>(xy)^3=\frac{810}{15}=54</math>. Plugging this into the second equation and dividing yields <math>x^3+y^3 = \frac{945}{54} = \frac{35}{2}</math>. Thus the sum required, as noted in Solution 1, is <math>54+\frac{35}{2} | + | Add three times the first equation to the second equation and factor to get <math>(xy)^3(x^3+3x^2y+3xy^2+y^3)=(xy)^3(x+y)^3=3375</math>. Taking the cube root yields <math>xy(x+y)=15</math>. Noting that the first equation is <math>(xy)^3\cdot(xy(x+y))=810</math>, we find that <math>(xy)^3=\frac{810}{15}=54</math>. Plugging this into the second equation and dividing yields <math>x^3+y^3 = \frac{945}{54} = \frac{35}{2}</math>. Thus the sum required, as noted in Solution 1, is <math>54+\frac{35}{2}\cdot2 = \boxed{089}</math>. |
==See also== | ==See also== | ||
{{AIME box|year=2015|n=II|num-b=13|num-a=15}} | {{AIME box|year=2015|n=II|num-b=13|num-a=15}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 15:55, 8 June 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 .
Solution 2
Factor the given equations as and , respectively. By the first equation, . Plugging this in to the second equation and simplifying yields . Now substitute . Solving the quadratic in , we get or As both of the original equations were symmetric in and , WLOG, let , so . Now plugging this in to either one of the equations, we get the solutions , . Now plugging into what we want, we get
Solution 3
Add three times the first equation to the second equation and factor to get . Taking the cube root yields . Noting that the first equation is , we find that . Plugging this into the second equation and dividing yields . Thus the sum required, as noted in Solution 1, is .
See also
2015 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.