Key Concepts and Formulas

• Arithmetic Progression (AP): For three terms $a, b, c$ in AP, the middle term is the arithmetic mean of the other two: $2b = a+c$.
• Geometric Progression (GP): For three terms $a, b, c$ in GP, the square of the middle term is the product of the other two: $b^2 = ac$.
• Algebraic Manipulation: Efficient use of substitution and simplification to solve systems of equations.

---

Step-by-Step Solution

Step 1: Translating the Arithmetic Progression Condition

We are given that $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in an Arithmetic Progression. Using the definition of AP, we have: $2 \left(\frac{1}{y}\right) = \frac{1}{x} + \frac{1}{z}$ To simplify this, we find a common denominator on the right side: $\frac{2}{y} = \frac{z+x}{xz} \quad \text{(Equation 1)}$ This equation establishes a relationship between $x, y,$ and $z$ based on the AP property.

Step 2: Translating the Geometric Progression Condition

We are given that $x, \sqrt{2} y, z$ are in a Geometric Progression. Applying the definition of GP: $(\sqrt{2}y)^2 = xz$ Simplifying the left side gives: $2y^2 = xz \quad \text{(Equation 2)}$ This equation provides another relationship between $x, y,$ and $z$ from the GP property.

Step 3: Combining AP and GP Conditions

We can combine Equation 1 and Equation 2 to find a simpler relationship between $x, y,$ and $z$. Substitute $xz = 2y^2$ (from Equation 2) into Equation 1: $\frac{2}{y} = \frac{x+z}{2y^2}$ Since we are given $0 < z < y < x$, we know that $y \neq 0$, so we can multiply both sides by $2y^2$: $2(2y^2) = y(x+z)$ $4y^2 = y(x+z)$ Again, since $y \neq 0$, we can divide both sides by $y$: $4y = x+z \quad \text{(Equation 3)}$ This is a significant simplification, showing that the sum of $x$ and $z$ is directly related to $y$.

Step 4: Using the Third Equation to Solve for $y$

We are given the equation $xy+yz+zx=\frac{3}{\sqrt{2}} x y z$. A strategic way to use this equation is to divide it by $xyz$ (which is permissible because $x, y, z$ are all positive real numbers, hence non-zero): $\frac{xy}{xyz} + \frac{yz}{xyz} + \frac{zx}{xyz} = \frac{3}{\sqrt{2}}$ This simplifies to: $\frac{1}{z} + \frac{1}{x} + \frac{1}{y} = \frac{3}{\sqrt{2}}$ From Step 1, we know that $\frac{1}{x} + \frac{1}{z} = \frac{2}{y}$. Substituting this into the equation: $\frac{2}{y} + \frac{1}{y} = \frac{3}{\sqrt{2}}$ $\frac{3}{y} = \frac{3}{\sqrt{2}}$ Equating the two sides, we find the value of $y$: $y = \sqrt{2}$ This step allowed us to find the specific value of $y$ by leveraging all the derived relationships.

Step 5: Calculating the Final Expression

We need to find the value of $3(x+y+z)^2$. From Equation 3, we have $x+z = 4y$. We can rewrite the sum $x+y+z$ as $(x+z)+y$. Substituting $x+z = 4y$: $x+y+z = 4y + y = 5y$ Now substitute this into the expression we want to evaluate: $3(x+y+z)^2 = 3(5y)^2$ $= 3(25y^2)$ $= 75y^2$ Finally, substitute the value of $y = \sqrt{2}$ that we found in Step 4: $= 75(\sqrt{2})^2$ $= 75(2)$ $= 150$

Common Mistakes & Tips

• Division by Zero: Always ensure variables are non-zero before dividing by them. The condition $0 < z < y < x$ guarantees this for $x, y, z$.
• Algebraic Simplification: Simplifying equations as early as possible, especially by combining terms or substituting known relationships, makes the problem much more manageable.
• Systematic Approach: Break down the problem into smaller, manageable steps by translating each given condition into an equation and then systematically combining them.

Summary

The problem was solved by first translating the given arithmetic and geometric progression conditions into algebraic equations. These equations were then combined to establish simpler relationships between the variables $x, y,$ and $z$, notably $x+z=4y$. The third given equation was then simplified using these relationships to solve for the value of $y$. Finally, the value of $y$ and the relationship $x+z=4y$ were used to calculate the required expression $3(x+y+z)^2$.

The final answer is $\boxed{150}$.