$$
K=(x-a)^{2}+(x-b)^{2}+(x-c)^{2}
$$

In the expression, $a, b, c$ are given real numbers, and $x$ is any real number. What is the smallest value of $K$?
In the expression $K=(x-a)^{2}+(x-b)^{2}+(x-c)^{2}$, after performing the squaring and rearranging, we obtain the following quadratic expression:

$$
K=3 x^{2}-2 x(a+b+c)+a^{2}+b^{2}+c^{2}
$$

Completing the square:

$$
\begin{gathered}
K=3\left(x^{2}-\frac{2}{3} x(a+b+c)+\frac{a^{2}+b^{2}+c^{2}}{3}\right)= \\
=3\left[\left(x-\frac{a+b+c}{3}\right)^{2}-\left(\frac{a+b+c}{3}\right)^{2}+\frac{a^{2}+b^{2}+c^{2}}{3}\right]
\end{gathered}
$$

The quadratic expression takes its minimum value when $x-\frac{a+b+c}{3}=0$, at which point, after substitution and rearrangement,

$$
K=\frac{(a-b)^{2}+(b-c)^{2}+(a-c)^{2}}{3}
$$

is the minimum value of $K$, which occurs at $x=\frac{a+b+c}{3}$.

Miklós Tirpák (Tapolca, Batsányi J. Gymnasium, 4th grade)