Problem III - 1

The non-zero real numbers $a$ and $b$ satisfy the equation

$$
\frac{a^{2} b^{2}}{a^{4}-2 b^{4}}=1
$$

Find, with justification, all the values taken by the expression

$$
\frac{a^{2}-b^{2}}{a^{2}+b^{2}}
$$
Solution

The first equality in the statement we write as

$$
a^{2} b^{2}=a^{4}-2 b^{4} \Leftrightarrow a^{4}-b^{2} a^{2}-2 b^{4}=0 \Leftrightarrow\left(a^{2}+b^{2}\right)\left(a^{2}-2 b^{2}\right)=0
$$

From this, either $a^{2}=-b^{2}$ (which is clearly impossible), or $a^{2}=2 b^{2}$.

In the latter case, substituting into the second expression, we obtain

$$
\frac{2 b^{2}-b^{2}}{2 b^{2}+b^{2}}=\frac{1}{3}
$$