The following expression is given:

$$
y=x^{2}-(8 a-2) x+15 a^{2}-2 a-7
$$

1. Determine the value of $a$ such that this expression is positive for all values of $a$.
2. Determine the value of $a$ such that the sum of the squares of the roots of the equation $y=0$ is 24.
1. The given expression can also be written as:

$$
y=[x-(4 a-1)]^{2}-\left(a^{2}-6 a+8\right)
$$

$y$ will be positive for all values of $x$ if

$$
a^{2}-6 a+8 \leqq 0
$$

or if

$$
(a-3)^{3}-1 \leqq 0
$$

This condition is satisfied if $a \geqq 2$ and $a \leqq 4$, that is, if

$$
4 \geqq a \geqq 2
$$

2. Based on the condition:

$$
x_{1}^{2}+x_{2}^{2}=24
$$

$$
x_{1}+x_{2}=8 a-2
$$

$$
x_{1} x_{2}=15 a^{2}-2 a-7
$$

If we subtract (1) from the square of (2) and then consider (3), we obtain

$$
17 a^{2}-14 a-3=0
$$

from which

$$
a_{1}=1, a_{2}=-\frac{3}{17}
$$

(Julius Lukhaub.)

The problem was also solved by: E. Freibauer, J. Kárf, Ö. Kornis, Gy. Krisztián, Gy. Perl, J. Weisz.