2007 Indonesia MO Problems/Problem 6
Problem
Find all triples of real numbers which satisfy the simultaneous equations
Solution
To start, since all three equations have a similar form, we can let to see if there are any solutions. Doing so results in Note that has complex solutions, so the solution where is .
Additionally, note that and are monotonically increasing functions, so is a monotonically increasing function. Thus, we can suspect that is the only solution. To prove this, we can use proof by contradiction.
Assume that . Because is a monotonically increasing function, , so and , so . By doing the same steps, we can show that and . However, that would mean that , which does not work, so there are no solutions where .
Similarly, assume that . Because is a monotonically increasing function, , so and , so . By doing the same steps, we can show that and . However, that would mean that , which does not work, so there are no solutions where .
Thus, we proved that is the only solution, and by substituting the value into the original equations, we get the only solution of .
See Also
2007 Indonesia MO (Problems) | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 7 |
All Indonesia MO Problems and Solutions |