What is the relationship between the parameters $a, b, c$ if the following equation has infinitely many solutions?

$$
\sqrt{x+a \sqrt{x}+b}=c
$$
$$
\sqrt{x+a \sqrt{x}+b}=c
$$

Equation (1) is squared. This is not an equivalent transformation, false roots may appear, which must be excluded from the solution set later. Grouping immediately, we get:

$$
(\sqrt{x})^{2}+a \sqrt{x}+\left(b-c^{2}\right)=0
$$

This is a quadratic equation in $\sqrt{x}$. We know that if the coefficient of the quadratic term in a quadratic equation is not zero, it has 0, 1, or 2 solutions. This is also the case here; (2) can have 0, 1, or 2 roots, including false roots. Therefore, equation (1) can never have infinitely many solutions.

# Kiss Attila (Budapest, Konyves Kalman Gymn., II. grade)

Note. Unfortunately, a $\sqrt{x}$ was missing from the problem statement. The original problem read as follows:

What relationship must exist between the parameters $a, b, c$ for the following equation to have infinitely many solutions?

$$
\sqrt{x+a \sqrt{x}+b}+\sqrt{x}=c
$$

Solution. After rearranging and squaring the equation,

$$
(a+2 c) \sqrt{x}=\left(c^{2}-b\right) .
$$

If (3) were satisfied for infinitely many $x$, (4) would also have to be satisfied for infinitely many $x$, which is only possible if

$$
a=-2 c \text { and } b=c^{2} \text {. }
$$

Substituting these into (3), we get $\sqrt{x-2 c \sqrt{x}+c^{2}}+\sqrt{x}=c$, or $|c-\sqrt{x}|+\sqrt{x}=c$, which means that $\sqrt{x} \leq c$ must hold for infinitely many $x$. This is precisely the case if

$$
0<c
$$

Thus, (5) and (6) must be satisfied for (3) to have infinitely many solutions, and if these are satisfied, (3) will have infinitely many solutions.

Bene Gyula (Miskolc, Foldes F. Gymn., II. grade)