9.3. Non-zero numbers $a, b$, and $c$ are such that the equalities $a^{2}(b+c-a)=b^{2}(a+c-b)=c^{2}(b+a-c)$ hold. What is the greatest value that the expression $\frac{2 b+3 c}{a}$ can take?
Answer: 5.

Solution: By equating the first and second expressions, after transformation, we get: $(a-b)\left(a^{2}+b^{2}-a c-b c\right)=0$. Similarly, we obtain the equalities $(b-c)\left(b^{2}+c^{2}-a b-a c\right)=0$ and $(a-c)\left(a^{2}+c^{2}-a b-c b\right)=0$.

We will prove that $a=b=c$.

Assume that $a=b \neq c$. Then from the second equality, we get: $0=b^{2}+c^{2}-a b-a c=b^{2}+c^{2}-b^{2}-b c=c^{2}-b c$, from which $b=c$ (since $c \neq 0$). Contradiction.

Thus, either the numbers are pairwise distinct, or they are all equal to each other.

Assume that the numbers are pairwise distinct, then $a^{2}+b^{2}-a c-b c=0$, $b^{2}+c^{2}-a b-a c=0$ and $a^{2}+c^{2}-a b-c b=0$. Adding these equalities, we get $(a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0$, from which $a=b=c$. Contradiction.

Thus, the only possible case is $a=b=c$. But then $\frac{2 b+3 c}{a}=\frac{2 a+3 a}{a}=5$.

Comment: The correct answer is obtained by considering a special case - 1 point.

It is claimed that $a=b=c$, but this statement is not proven - no points are added.

9.4-1. It is known that the quadratic equation $a x^{2}+b x+c=0$ has no solutions. Is it true that the quadratic equation $(2 a+c) x^{2}+3 b x+(a+2 c)=0$ also has no solutions?

Answer: True.

Solution: Note that if the quadratic equation $a x^{2}+b x+c=0$ has no solutions, then the quadratic equation $c x^{2}+b x+a=0$ also has no solutions, as these equations have the same discriminant. Moreover, if the quadratic equation $a x^{2}+b x+c=0$ has no solutions, then the signs of the coefficients $a$ and $c$ will be the same (otherwise, the discriminant would be non-negative). This means that the graphs of the quadratic functions $f(x)=a x^{2}+b x+c$ and $g(x)=c x^{2}+b x+a$ either both lie above the $O x$ axis or both lie below the $O x$ axis. But then the graph of the sum $h(x)=2 f(x)+g(x)=(2 a+c) x^{2}+3 b x+(a+2 c)$ also does not intersect the $O x$ axis.

Comment: The correct answer without justification - 0 points.

9.4-2. It is known that the quadratic equation $a x^{2}+b x+c=0$ has no solutions. Is it true that the quadratic equation $(a+3 c) x^{2}+4 b x+(3 a+c)=0$ also has no solutions?

Answer: True.

Solution: Note that if the quadratic equation $a x^{2}+b x+c=0$ has no solutions, then the quadratic equation $c x^{2}+b x+a=0$ also has no solutions, as these equations have the same discriminant. Moreover, if the quadratic equation $a x^{2}+b x+c=0$ has no solutions, then the signs of the coefficients $a$ and $c$ will be the same (otherwise, the discriminant would be non-negative). This means that the graphs of the quadratic functions $f(x)=a x^{2}+b x+c$ and $g(x)=c x^{2}+b x+a$ either both lie above the $O x$ axis or both lie below the $O x$ axis. But then the graph of the sum $h(x)=f(x)+3 g(x)=(a+3 c) x^{2}+4 b x+(3 a+c)$ also does not intersect the $O x$ axis.

Comment: The correct answer without justification - 0 points.