For the equations $a x^{2}+b x+c=0, \alpha y^{2}+\beta y+\gamma=0$, let $a \alpha \neq 0, b^{2}-4 a c \geq 0, \beta^{2}-4 \alpha \gamma \geq 0$, and the roots satisfy $x_{1} \leq x_{2}, y_{1} \leq y_{2}$. Write down the quadratic equations in $z$ and $u$ whose roots are the following pairs:

$$
z_{1}=x_{1}+y_{1}, \quad z_{2}=x_{2}+y_{2} ; \quad u_{1}=x_{1}+y_{2}, \quad u_{2}=x_{2}+y_{1}
$$
Due to the assumptions, the given equations are indeed quadratic, and their roots are real. Let's briefly state

$$
b^{2}-4 a c=d^{2} \quad \text { and } \quad \beta^{2}-4 \alpha \gamma=\delta^{2}
$$

We also assume that $a>0, \alpha>0$, which we can achieve by multiplying the given equations by (-1) if necessary. Then, for example, $x_{1}=(-b-d) / 2 a, y_{2}=(-\beta+\delta) / 2$, where $d$ and $\delta$ represent the non-negative square roots of the discriminants.

A quadratic equation that meets the requirements, based on the known relationships between the roots and coefficients, is

$$
z^{2}-\left(z_{1}+z_{2}\right) z+z_{1} z_{2}=0
$$

or

$$
u^{2}-\left(u_{1}+u_{2}\right) u+u_{1} u_{2}=0
$$

The coefficient of the linear term, according to (1), and applying the stated relationships to the given equations, in case (2) is

$$
-\left(z_{1}+z_{2}\right)=-\left(x_{1}+x_{2}\right)-\left(y_{1}+y_{2}\right)=\frac{b}{a}+\frac{\beta}{\alpha}=\frac{1}{a \alpha}(a \beta+\alpha b)
$$

and in case (3) it is the same. The constant term is

$$
\begin{aligned}
z_{1} z_{2}=x_{1} x_{2}+y_{1} y_{2}+x_{1} y_{2}+x_{2} y_{1} & =\frac{c}{a}+\frac{\gamma}{\alpha}+\frac{1}{4 a \alpha}[(-b-d)(-\beta+\delta)+ \\
+(-b+d)(-\beta-\delta)] & =\frac{2 a \gamma+2 \alpha c+b \beta-d \delta}{2 a \alpha}
\end{aligned}
$$

and similarly,

$$
u_{1} u_{2}=x_{1} x_{2}+y_{1} y_{2}+x_{1} y_{1}+x_{2} y_{2}=\frac{2 a \gamma+2 \alpha c+b \beta+d \delta}{2 a \alpha}
$$

Based on these, a suitable equation, after removing the fractions, is:

$$
\begin{aligned}
& 2 a \alpha z^{2}+2(a \beta+\alpha b) z+(2 a \gamma+2 \alpha c+b \beta-d \delta)=0 \\
& 2 a \alpha u^{2}+2(a \beta+\alpha b) u+(2 a \gamma+2 \alpha c+b \beta+d \delta)=0
\end{aligned}
$$

István Fodor (Budapest, Kölcsey F. Gymnasium III. grade)