MLD

Let $x, y, z$ be real numbers, satisfying the relations

$$
\left\{\begin{array}{l}
x \geq 20 \\
y \geq 40 \\
z \geq 1675 \\
x+y+z=2015
\end{array}\right.
$$

Find the greatest value of the product $P=x \cdot y \cdot z$.
By virtue of \( z \geq 1675 \) we have

\[
y+z<2015 \Leftrightarrow y<2015-z \leq 2015-1675<1675
\]

It follows that \((1675-y) \cdot (1675-z) \leq 0 \Leftrightarrow y \cdot z \leq 1675 \cdot (y+z-1675)\).

By using the inequality \( u \cdot v \leq \left(\frac{u+v}{2}\right)^2 \) for all real numbers \( u, v \) we obtain

\[
\begin{gathered}
P = x \cdot y \cdot z \leq 1675 \cdot x \cdot (y+z-1675) \leq 1675 \cdot \left(\frac{x+y+z-1675}{2}\right)^2 = \\
1675 \cdot \left(\frac{2015-1675}{2}\right)^2 = 1675 \cdot 170^2 = 48407500
\end{gathered}
\]

\[
\text{We have } P = x \cdot y \cdot z = 48407500 \Leftrightarrow \left\{\begin{array}{l}
x + y + z = 2015, \\
z = 1675, \\
x = y + z - 1675
\end{array} \Leftrightarrow \left\{\begin{array}{l}
x = 170 \\
y = 170 \\
z = 1675
\end{array}\right.\right.
\]

So, the greatest value of the product is \( P = x \cdot y \cdot z = 48407500 \).

## Solution 2:

Let \( S = \{(x, y, z) \mid x \geq 20, y \geq 40, z \geq 1675, x + y + z = 2015\} \) and \( \Pi = \{|x \cdot y \cdot z| (x, y, z) \in S\} \). We have to find the biggest element of \( \Pi \). By using the given inequalities we obtain:

\[
\left\{\begin{array}{l}
20 \leq x \leq 300 \\
40 \leq y \leq 320 \\
1675 \leq z \leq 1955 \\
y < 1000 < z
\end{array}\right.
\]

Let \( z = 1675 + d \). Since \( x \leq 300 \) so \((1675 + d) \cdot x = 1675 x + d x \leq 1675 x + 1675 d = 1675 \cdot (x + d)\). That means that if \((x, y, 1675 + d) \in S\) then \((x + d, y, 1675) \in S\), and \( x \cdot y \cdot (1675 + d) \leq (x + d) \cdot y \cdot 1675 \). Therefore \( z = 1675 \) must be for the greatest product.

Furthermore, \( x \cdot y \leq \left(\frac{x + y}{2}\right)^2 = \left(\frac{2015 - 1675}{2}\right)^2 = \left(\frac{340}{2}\right)^2 = 170^2 \). Since \((170, 170, 1675) \in S\) that means that the biggest element of \( \Pi \) is \( 170 \cdot 170 \cdot 1675 = 48407500 \).