2015 AIME II Problems/Problem 14
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 .
Note that since the value we want to find is , we can convert into an expression in terms of , since from the second equation which is , we see that and thus the value is Since we've already found we substitute and find the answer to be 89.
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 .
Solution 4
As with the other solutions, factor. But this time, let and . Then . Notice that . Now, if we divide the second equation by the first one, we get ; then . Therefore, . Substituting into in equation 2 gives us ; we are looking for . Finding , we get . Substituting into the first equation, we get . Our final answer is .
Solution 5
Factor the given equations as: We note that these expressions (as well as the desired expression) can be written exclusively in terms of and . We make the substitution and (for sum and product, respectively).
We see that shows up in both equations, so we can eliminate it and find , after which we can get from the first equation. If you rewrite the desired expression using and , it becomes clear that you don't need to actually find the values of and , but I will do so for the sake of completion.
The desired expression can be written as:
Plugging in and , we get:
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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 |
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