## Task 1 - 261221

Determine all triples of real numbers $(x ; y ; z)$ that are solutions to the following system of equations (1), (2), (3):

$$
\begin{aligned}
x \cdot y & =2 \\
x \cdot z & =3 \\
x^{2}+y^{2} & =5
\end{aligned}
$$
I. If a triple $(x ; y ; z)$ of real numbers is a solution to the system of equations (1), (2), (3), then: From (1), $y \neq 0$, so from (1) we have $x=\frac{2}{y}$. Substituting this into (3) yields $x^{2}+\frac{4}{x^{2}}=5$, thus

$$
x^{4}-5 x^{2}+4=0
$$

Therefore, the number $u=x^{2} \quad$ (5) satisfies the equation $u^{2}-5 u+4=0 \quad$ (6).

From (6), it follows that $u$ is one of the numbers

$$
u_{1,2}=\frac{1}{2}(5 \pm \sqrt{25-16}) \quad \Rightarrow u_{1}=4, \quad u_{2}=1
$$

Thus, $x$ is, according to (5), one of the numbers

$$
x_{1}=2 ; \quad x_{2}=-2 ; \quad x_{3}=1 ; \quad x_{4}=-1
$$

From (1), the corresponding value for $y$ is

$$
y_{1}=1 ; \quad y_{2}=-1 ; \quad y_{3}=2 ; \quad y_{4}=-2
$$

and from (2), the corresponding value for $z$ is

$$
z_{1}=\frac{3}{2} ; \quad z_{2}=-\frac{3}{2} ; \quad z_{3}=3 ; \quad z_{4}=-3
$$

Therefore, only the triples

$$
\left(2 ; 1 ; \frac{3}{2}\right) ; \quad\left(-2 ;-1 ;-\frac{3}{2}\right) ; \quad(1 ; 2 ; 3) ; \quad(-1 ;-2 ;-3)
$$

can satisfy the system of equations (1), (2), (3). They do satisfy it, as can be verified by substitution.

Adapted from $[5]$

2. Solution:

If we consider the equation "(3)+2 (1)", we obtain $9=x^{2}+2 x y+y^{2}=(x+y)^{2}$, thus $x+y= \pm 3$. Similarly, from "(3)+2 (1)" we get the equation $1=x^{2}-2 x y+y^{2}=(x-y)^{2}$, thus $x-y= \pm 1$.

We perform a complete case analysis based on the signs of $x+y$ and $x-y$:

Case 1: $x+y=3$.

Case 1.1: $x-y=1$. It follows that $(x, y, z)=\left(2,1, \frac{3}{2}\right)$.

Case 1.2: $x-y=-1$. It follows that $(x, y, z)=(1,2,3)$.

Case 2: $x+y=-3$.

Case 2.1: $x-y=1$. It follows that $(x, y, z)=(-1,-2,-3)$.

Case 2.2: $x-y=-1$. It follows that $(x, y, z)=\left(-2,-1,-\frac{3}{2}\right)$.

Thus, exactly the four mentioned triples solve the system of equations, as confirmed by substitution.