Let's determine the value of the parameter $p$ such that for the roots $x_{1}$ and $x_{2}$ of the equation

$$
x^{2}+p x+p-1=0
$$

the following condition is satisfied:

$$
x_{1}^{2}+x_{1}^{3}=-\left(x_{2}^{2}+x_{2}^{3}\right)
$$
I. solution. By separating the terms containing $p$, the left side of (1) can be transformed into a product:

$$
\left(x^{2}-1\right)+p(x+1)=(x+1)(x-1+p)
$$

Accordingly, the roots are: $x_{1}=-1, x_{2}=1-p$. Substituting $x_{1}$ into (2), it can be rewritten as:

$$
x_{2}^{2}\left(1+x_{2}\right)=0
$$

This holds if and only if either

$$
\begin{gathered}
x_{2}=0, \quad \text { i.e., } \quad 1-p=0 \text {, hence } \quad p=1, \quad \text { or } \\
1+x_{2}=0, \quad \text { i.e., } \quad 2-p=0, \quad p=2
\end{gathered}
$$

Tamás Kunszenti (Budapest, Petőfi S. g. IV. o. t.)

II. solution. We can reach the goal without writing down the roots by using the relationships between the roots and the coefficients. Reducing (2) to 0,

$$
\left(x_{1}^{2}+x_{2}^{2}\right)+\left(x_{1}^{3}+x_{2}^{3}\right)=\left(x_{1}+x_{2}\right)^{2}-2 x_{1} x_{2}\left(x_{1}+x_{2}\right)^{3}-3 x_{1} x_{2}\left(x_{1}+x_{2}\right)=0
$$

Substituting the expressions $x_{1}+x_{2}=-p, x_{1} x_{2}=p-1$ derived from (1) into this, we get an equation for $p$:

$$
p^{2}-2(p-1)-p^{3}+3 p(p-1)=-p^{3}+4 p^{2}-5 p+2=0
$$

Noticing that the sum of the coefficients is 0, we see that one root of (4) is $p=1$, so $p-1$ can be factored out on the left side:

$$
(p-1)\left(-p^{2}+3 p-2\right)=0
$$

And since the roots of the polynomial in the second parenthesis are 1 and 2, our equation becomes

$$
-(p-1)^{2}(p-2)=0
$$

which holds for $p=1$ and $p=2$.

Miklós Verdes (Pannonhalma, Bencés g. III. o. t.)

Remarks. 1. From (3), it is clear that with the determined values of $p$ - and only with these - for every positive integer $n$, the more general requirement

$$
x_{1}^{n}+x_{1}^{n+1}=-\left(x_{2}^{n}+x_{2}^{n+1}\right)
$$

is satisfied.

Andor Surányi (Budapest, Madách I. g. IV. o. t.)

2. (5) can also be obtained from (4) by factoring out $p^{2}-p^{3}=-p^{2}(p-1)$.