2017 Indonesia MO Problems/Problem 4
Problem
Determine all pairs of distinct real numbers such that both of the following are true:
Solution
Let and resulting in the below system. Substitution and factoring of difference of squares results in By the Zero Product Property, either or If and then However, a quick check reveals this to be an extraneous solution.
Thus, If then and if then Now assume Because we have Since is positive, and must have the same sign.
Note that If are negative, then the sum is negative since each term is negative, so must both be positive. That means so
Since we have However, that can not happen, so there are no more solutions.
This means that the only solutions are
See Also
2017 Indonesia MO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 5 |
All Indonesia MO Problems and Solutions |