# Problem 1.

There are three electric generators with powers $x_{1}, x_{2}, x_{3}$ less than 1 MW. When analyzing the power system with such generators, it was found that for a certain process to be carried out, the condition

$$
2\left(x_{1}+x_{2}+x_{3}\right)+4 x_{1} x_{2} x_{3}=3\left(x_{1} x_{2}+x_{1} x_{3}+x_{2} x_{3}\right)+1 .
$$

What is the maximum combined power of all three generators when this condition is met?
# Solution.

Let's rewrite the equality from the condition as follows:

$$
\begin{gathered}
4 x_{1} x_{2} x_{3}-4\left(x_{1} x_{2}+x_{1} x_{3}+x_{2} x_{3}\right)+4\left(x_{1} x_{2} x_{3}\right)-4= \\
\quad=-\left(x_{1} x_{2}+x_{1} x_{3}+x_{2} x_{3}\right)+2\left(x_{1}+x_{2}+x_{3}\right)-3 \\
4\left(x_{1}-1\right)\left(x_{2}-1\right)\left(x_{3}-1\right)=-\left(x_{1}-1\right)\left(x_{2}-1\right)-\left(x_{1}-1\right)\left(x_{3}-1\right)-\left(x_{2}-1\right)\left(x_{3}-1\right) \\
4=1 /\left(1-x_{1}\right)+1 /\left(1-x_{2}\right)+1 /\left(1-x_{3}\right)
\end{gathered}
$$

Consider the function \( f(x) = 1 / (1 - x) \) on the interval \((-\infty, 1)\). The function is concave (visible on the graph), so

$$
4 = f\left(x_{1}\right) + f\left(x_{2}\right) + f\left(x_{3}\right) \geq 3 f\left(\left(x_{1} + x_{2} + x_{3}\right) / 3\right) = \frac{9}{3 - \left(x_{1} + x_{2} + x_{3}\right)}
$$

(A function \( f_{1}(x) \) is linear (neither convex nor concave) on the interval \( X \) if and only if for any \( x_{1}, x_{2}, x_{3} \in X \) the equality holds

$$
f_{1}\left(\left(x_{1} + x_{2} + x_{3}\right) / 3\right) = \left(f_{1}\left(x_{1}\right) + f_{1}\left(x_{2}\right) + f_{1}\left(x_{3}\right)\right) / 3
$$

A function \( f_{2}(x) \) is concave on \( X \) if and only if

$$
\forall x_{1}, x_{2}, x_{3} \in X \quad f_{2}\left(\left(x_{1} + x_{2} + x_{3}\right) / 3\right) \leq \left(f_{2}\left(x_{1}\right) + f_{2}\left(x_{2}\right) + f_{2}\left(x_{3}\right)\right) / 3
$$

A function \( f_{3}(x) \) is convex on \( X \) if and only if

$$
\forall x_{1}, x_{2}, x_{3} \in X \quad f_{3}\left(\left(x_{1} + x_{2} + x_{3}\right) / 3\right) \leq \left(f_{3}\left(x_{1}\right) + f_{3}\left(x_{2}\right) + f_{3}\left(x_{3}\right)\right) / 3
$$

This can be easily verified by plotting the graphs of functions \( f_{1}, f_{2}, f_{3} \) in the same coordinate system with the same \( x_{1}, x_{2}, x_{3} \) and values \( f_{1}\left(x_{j}\right) = f\left(x_{j}\right) = f_{3}\left(x_{j}\right) \), \( j=1,2,3 \).

From this, we obtain the required inequality

$$
x_{1} + x_{2} + x_{3} \leq 3 / 4
$$

## Answer: \( 3 / 4 \).

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