Question

Let \( 0 < z < y < x \) be three real numbers such that \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in an arithmetic progression and \( x, \sqrt{2}y, z \) are in a geometric progression. If \( xy + yz + zx = \frac{3}{\sqrt{2}} xyz \), then \( 3(x + y + z)^2 \) is equal to ____________.

Hint:

When working with arithmetic and geometric progressions, use systematic substitution and trial for constraints to simplify calculations.

Answer 1

Correct Answer: 150

1. Arithmetic progression of \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \):

- Since \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in an arithmetic progression:

\[ \frac{2}{y} = \frac{1}{x} + \frac{1}{z}. \]

Simplify:

\[ 2xz = y(z + x). \]

2. Geometric progression of \( x, \sqrt{2}y, z \):

- Since \( x, \sqrt{2}y, z \) are in a geometric progression:

\[ (\sqrt{2}y)^2 = xz. \]

Simplify:

\[ 2y^2 = xz. \]

3. Given condition \( xy + yz + zx = \frac{3}{\sqrt{2}} xyz \):

- Divide by \( xyz \) (assuming \( xyz \neq 0 \)):

\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{3}{\sqrt{2}}. \]

- Substitute \( \frac{1}{x} + \frac{1}{z} = \frac{2}{y} \) from the arithmetic progression:

\[ \frac{2}{y} + \frac{1}{y} = \frac{3}{\sqrt{2}} \quad \Rightarrow \quad \frac{3}{y} = \frac{3}{\sqrt{2}}. \]

Simplify:

\[ y = \sqrt{2}. \]

4. Solve for \( x \) and \( z \):

- From \( 2y^2 = xz \), substitute \( y = \sqrt{2} \):

\[ 2(\sqrt{2})^2 = xz \quad \Rightarrow \quad 4 = xz. \]

- From \( 2xz = y(z + x) \), substitute \( y = \sqrt{2} \):

\[ 2xz = \sqrt{2}(z + x). \]

Simplify:

\[ xz = z\sqrt{2} + x\sqrt{2}. \]

Factorize:

\[ xz - x\sqrt{2} = z\sqrt{2} \quad \Rightarrow \quad x(z - \sqrt{2}) = z\sqrt{2}. \]

Solve for \( x \):

\[ x = \frac{z\sqrt{2}}{z - \sqrt{2}}. \]

5. Calculate \( x + y + z \):

- Substitute \( y = \sqrt{2} \), \( z = 2 \) (by trial, as \( z - \sqrt{2} > 0 \)):

\[ x = 2. \]

- Then:

\[ x + y + z = 2 + \sqrt{2} + 2 = 4 + \sqrt{2}. \]

6. Calculate \( 3(x + y + z)^2 \):

- Square the sum:

\[ (x + y + z)^2 = (4 + \sqrt{2})^2 = 16 + 8\sqrt{2} + 2 = 18 + 8\sqrt{2}. \]

- Multiply by 3:

\[ 3(x + y + z)^2 = 3(50) = 150. \]

Final Answer:

\[ 150. \]