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Ukrainian National Mathematical Olympiad

Algebra Difficulty 5.3 AIME, harder Prove it Ukraine

Find possible positive solutions $(x, y, z)$ for the given system of equations:
$\begin{cases} x + y^2 = 2z^3, \\ y + z^2 = 2x^3, \\ z + x^2 = 2y^3. \end{cases}$

Solution

Without loss of generality let $z \ge \max\{x, y\}$ . Consider the following cases.

Case 1. $z > 1$ . Then due to the first equation $x + y^2 \le z + z^2 < 2z^3$ – contradiction.

Case 2. $z = 1$ . Due to the first equation, $2 = x + y^2$ .
Since $\max\{x, y\} \le 1$ , it is possible only if $x = y = 1$ . It is easy to check that $(1; 1; 1)$ is a solution.

Case 3. $z < 1$ . Then for the smallest variable, say $x$ , $x < 1$ holds, then by the second equation:
$x + y^2 \le z + z^2 < 2z^3$ – contradiction.

Similar contradiction holds if $y$ is the smallest variable.

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