2006 Alabama ARML TST Problems/Problem 14
Problem
Find the real solution to the system of equations
Solution
Note that
So, we need to find that satisfies . Even though there are three solutions, for simplicity, lets assume that one exists in the first quadrant in the complex plane, and try to find that one.
Note that .
Thus, .
Therefore, . Note that and are the only positive integer solutions to . (Even though the solution may be non-integral, this is a good place to start.)
However, letting , [] and bounding yields:
.
. (Since is obviously in the 2nd quadrant).
.
.
Thus, . So, the only remaining positive integer solution is .
As a quick check:
Thus, the solution is .
Note that and
Thus, and are also solutions.
See also
2006 Alabama ARML TST (Problems) | ||
Preceded by: Problem 13 |
Followed by: 15 | |
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