Iberoamerican integer system of equations, 2018
Problem. For each integer , find all integer solutions of the following system of equations:
Origin: Iberoamerican Mathematical Olympiad 2018, problem 1.
Solution. First, notice that for any , thus the right sides of all equations are non-negative. Therefore, so are the left sides, and so all , .
Now, let's consider which is the smallest among , \ldots, . Then we have This can only be true either when , or when and . Consider these two cases separately.
- . Then . Then . Then all . Finally, we can check directly that this result satisfies the original equations.
- and . Then . Without loss of generality, let's assume that . Then we have and the two equations are and . Therefore, . Finally, we can check directly that this result satisfies the original equations.
To summarize, for , there are two solutions: and ; for , there is only one solution: for .