$$
x^{2}+p x+q=0 \quad \text { and } \quad x^{2}+p^{\prime} x+q^{\prime}=0
$$

are equations with roots $\alpha, \beta, \alpha^{\prime}$ and $\beta^{\prime}$. Express

$$
\left(\alpha-\alpha^{\prime}\right)\left(\alpha-\beta^{\prime}\right)\left(\beta-\alpha^{\prime}\right)\left(\beta-\beta^{\prime}\right)
$$

as a function of the coefficients $p, q, p^{\prime}, q^{\prime}$.
$$
\begin{gathered}
\quad\left(\alpha-\alpha^{\prime}\right)\left(\alpha-\beta^{\prime}\right)\left(\beta-\alpha^{\prime}\right)\left(\beta-\beta^{\prime}\right)=\left[\alpha^{2}-\alpha\left(\alpha^{\prime}+\beta^{\prime}\right)+\alpha^{\prime} \beta^{\prime}\right]\left[\beta^{2}-\beta\left(\alpha^{\prime}+\beta^{\prime}\right)+\alpha^{\prime} \beta^{\prime}\right]= \\
=\left[\alpha^{2}+\alpha p^{\prime}+q^{\prime}\right]\left[\beta^{2}+\beta p^{\prime}+q^{\prime}\right]=\alpha^{2} \beta^{2}+\alpha \beta^{2} p^{\prime}+\beta^{2} q^{\prime}+\alpha^{2} \beta p^{\prime}+\alpha \beta p^{\prime 2}+\beta p^{\prime} q+\alpha^{2} q^{\prime}+\alpha p^{\prime} q^{\prime}+q^{\prime 2}= \\
=q^{2}+p^{\prime} q\left[\alpha+\beta+p^{\prime}\right]+p^{\prime} q^{\prime}[\alpha+\beta]+q^{\prime}\left[\alpha^{2}+\beta^{2}\right]+q^{\prime 2}=q^{2}+p^{\prime} q\left[p^{\prime}-p\right]-p p^{\prime} q^{\prime}+q^{\prime}\left[p^{2}-2 q\right]+q^{\prime 2}= \\
=q^{2}-2 q q^{\prime}+q^{\prime 2}+p^{\prime} q\left[p^{\prime}-p\right]-p q^{\prime}\left[p^{\prime}-p\right]=\left[q-q^{\prime}\right]^{2}+\left[p-p^{\prime}\right]\left[q^{\prime} p-p^{\prime} q\right] .
\end{gathered}
$$

(József Sonnenfeld, Budapest.)

This expression can also be written in the following form:

$$
\left|\begin{array}{cc}
q & 1 \\
q^{\prime} & 1
\end{array}\right|^{2}+\left|\begin{array}{cc}
p & q \\
p^{\prime} & q^{\prime}
\end{array}\right| \cdot\left|\begin{array}{cc}
p & 1 \\
p^{\prime} & 1
\end{array}\right| .
$$

(Gyula Jánosy, Budapest.)

The problem was also solved by: E. Ádámffy, L. Bánó, I. Dömény, M. Fekete, H. Friedländer, A. Haar, Z. Harsányi, J. Hermann, J. Kiss, Gy. Krampera, I. Kürti, F. Magyari, P. Messer, A. Nagy, P. Rássy, M. Riesz, J. Rosenberg, I. Schöffer, Gy. Söpkéz, the V. district gymnasium mathematical circle.