10.3.1. (12 points) The curve given by the equation $y=2^{p} x^{2}+5 p x-2^{p^{2}}$ intersects the $O x$ axis at points $A$ and $B$, and the $O y$ axis at point $C$. Find the sum of all values of the parameter $p$ for which the center of the circle circumscribed around triangle $A B C$ lies on the $O x$ axis.
Answer: -1.

Solution. Let $x_{1}$ and $x_{2}$ be the roots of the quadratic trinomial $a x^{2}+b x+c (c \neq 0)$, the graph of which intersects the $O x$ axis at points $A$ and $B$, and the $O y$ axis at point $C$. Then $A\left(x_{1} ; 0\right), B\left(x_{2} ; 0\right), C(0 ; c)$. The fact that the center of the circle circumscribed around triangle $A B C$ lies on the $O x$ axis is equivalent to the angle $A C B$ being a right angle. It is not difficult to find that the slope of the line $A C$ is $k_{1}=-\frac{c}{x_{1}}$, and the slope of the line $B C$ is $k_{2}=-\frac{c}{x_{2}}$. The perpendicularity of these lines is equivalent to the condition $k_{1} \cdot k_{2}=-1$, that is, $x_{1} x_{2}=-c^{2}$, which, taking into account Vieta's theorem, can be rewritten as $a c=-1$.

For the trinomial from the problem statement, the condition $a c=-1$ becomes $2^{p+p^{2}}=1$, from which $p=0$, $p=-1$.