What relationship exists between the coefficients of the equation

$$
a x^{3}+b x^{2}+c x+d=0, \quad a d \neq 0
$$

if one of the roots is equal to the harmonic mean of the other two?
I. Solution. The harmonic mean of two numbers, $u$ and $v$, denoted by $h$, is the reciprocal of the arithmetic mean of their reciprocals:

$$
\frac{1}{h}=\frac{1}{2}\left(\frac{1}{u}+\frac{1}{v}\right), \quad \text { hence } \quad h=\frac{2 u v}{u+v}
$$

which only makes sense if $u, v$, and $u+v$ are all non-zero.

Let the roots of (1) be $x_{1}, x_{2}, x_{3}$, and let $x_{2}$ play the prominent role, then

$$
x_{2}=\frac{2 x_{1} x_{3}}{x_{1}+x_{3}}
$$

It is well known that the following relationships hold between the three roots and the coefficients of (1):

$$
x_{1}+x_{2}+x_{3}=-\frac{b}{a}, \quad x_{1} x_{2}+x_{1} x_{3}+x_{2} x_{3}=\frac{c}{a}, \quad x_{1} x_{2} x_{3}=-\frac{d}{a}
$$

Substituting (2) into these, we have three equations in the unknowns $x_{1}, x_{3}$, which can only be satisfied if there is a certain relationship among the right-hand sides. The task is to find this relationship, which we can do by solving for $x_{1}$ and $x_{3}$ from any two equations and substituting them into the remaining one.

$$
\begin{gathered}
x_{1}+x_{3}+\frac{2 x_{1} x_{3}}{x_{1}+x_{3}}=-\frac{b}{a}, \\
x_{1} x_{3}+\left(x_{1}+x_{3}\right) x_{2}=3 x_{1} x_{3}=\frac{c}{a} \\
\frac{2 x_{1}^{2} x_{3}^{2}}{x_{1}+x_{3}}=-\frac{d}{a}
\end{gathered}
$$

From the second equation, we see that $c \neq 0$ must also hold, since $d \neq 0$ implies that no root is zero. From (4), the last equation becomes

$$
x_{1}+x_{3}=-\frac{2 a}{d}\left(x_{1} x_{3}\right)^{2}=-\frac{2 a}{d} \cdot\left(\frac{c}{3 a}\right)^{2}=-\frac{2 c^{2}}{9 a d}
$$

which, together with (4), forms a system of two equations in the unknowns $x_{1}, x_{3}$.

However, we do not need separate expressions for $x_{1}$ and $x_{3}$, because in (3) we need symmetric functions of $x_{1}$ and $x_{3}$. Thus, from (3), using (5) and (4), the desired relationship is:

$$
-\frac{2 c^{2}}{9 a d}+\left(\frac{2 c}{3 a}\right) \cdot\left(-\frac{9 a d}{2 c^{2}}\right)=-\frac{2 c^{2}}{9 a d}-\frac{3 d}{c}=-\frac{b}{a}
$$

or, rearranging in the usual way,

$$
27 a d^{2}-9 b c d+2 c^{3}=0
$$

provided $a d \neq 0$. (The condition $c \neq 0$ found along the way does not need to be repeated, because from (6),

$$
\left.c=\frac{27 a d^{2}}{9 b d-2 c^{2}} \neq 0 .\right)
$$

II. Solution. Let the roots of (1) be $x_{1}, x_{2}, x_{3}$. Since $a \neq 0$, the equation is indeed of the third degree, and since $d \neq 0$, no root can be zero. If one of the roots is the harmonic mean of the other two, then the reciprocal of this root is the arithmetic mean of the reciprocals of the other two. The reciprocals of these roots are the roots of the equation

$$
d x^{3}+c x^{2}+b x+a=0
$$

since, for example, if $x_{1}$ is a root of (1), it means that

$$
a x_{1}^{3}+b x_{1}^{2}+c x_{1}+d=0
$$

and dividing by $x_{1}^{3}$ (which is not zero), we get that $\frac{1}{x_{1}}$ is a root of (7).

Denoting the roots of (7) by $y_{1}, y_{2}, y_{3}$, the relationships between the roots and coefficients are

$$
y_{1}+y_{2}+y_{3}=-\frac{c}{d}, \quad y_{1} y_{2}+y_{2} y_{3}+y_{3} y_{1}=\frac{b}{d}, \quad y_{1} y_{2} y_{3}=-\frac{a}{d}
$$

One of these roots will be the arithmetic mean of the other two if and only if the product

$$
S=\left(2 y_{1}-y_{2}-y_{3}\right)\left(2 y_{2}-y_{3}-y_{1}\right)\left(2 y_{3}-y_{1}-y_{2}\right)
$$

is zero. From (8), $S$ can be expressed in terms of the coefficients of the equation:

$$
\begin{gathered}
S=\left(3 y_{1}+\frac{c}{d}\right)\left(3 y_{2}+\frac{c}{d}\right)\left(3 y_{3}+\frac{c}{d}\right)= \\
=27 y_{1} y_{2} y_{3}+9\left(y_{1} y_{2}+y_{2} y_{3}+y_{3} y_{1}\right) \frac{c}{d}+3\left(y_{1}+y_{2}+y_{3}\right) \frac{c^{2}}{d^{2}}+\frac{c^{3}}{d^{3}}= \\
=-\frac{27 a}{d}+\frac{9 b c}{d^{2}}-\frac{3 c^{3}}{d^{3}}+\frac{c^{3}}{d^{3}}=\frac{1}{d^{3}}\left(-27 a d^{2}+9 b c d-2 c^{3}\right)
\end{gathered}
$$

Thus, $S$ is zero if and only if (6) holds, meaning that (6) is a necessary and sufficient condition for one of the roots of (1) to be the harmonic mean of the other two.