## Task 1 - 321221

Determine for each integer $k$ all pairs $(x ; y)$ of integers that satisfy the system of the following two equations (1), (2):

$$
\begin{aligned}
& x^{2}+k \cdot y^{2}=4 \\
& k \cdot x^{2}-y^{2}=2
\end{aligned}
$$
If one multiplies (1) by $k$ and subtracts (2), then one obtains

$$
\begin{gathered}
\left(k^{2}+1\right) y^{2}=4 k-2 \\
y^{2}=\frac{4 k-2}{k^{2}+1}
\end{gathered}
$$

It must be $k \geq 1$, otherwise $y$ would not be real. Moreover, $y$ cannot be zero. Therefore, it must hold that:

$$
\begin{aligned}
& y^{2}=\frac{4 k-2}{k^{2}+1} \geq 1 \\
& 4 k-2 \geq k^{2}+1 \\
& k^{2}-4 k+3 \leq 0
\end{aligned}
$$

This implies $1 \leq k \leq 3$. The possible solutions can thus be quickly tested, and it turns out that only for $k=3$ integer solutions are obtained, namely $x^{2}=y^{2}=1$. The solution pairs are therefore $(1 ; 1),(1 ;-1),(-1 ; 1)$ and $(-1 ;-1)$.