If x, y, z are distinct positive numbers such that `x+(1)/(y)=y+(1)/(z)=z+(1)/(x)`, then the value of xyz is __________

Answer

To solve the problem, we start with the given equation:

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

Let's denote this common value as \( k \). Therefore, we can write:

1. \( x + \frac{1}{y} = k \)
2. \( y + \frac{1}{z} = k \)
3. \( z + \frac{1}{x} = k \)

From these equations, we can express \( x, y, z \) in terms of \( k \):

### Step 1: Rearranging the equations

From the first equation:

\[ x = k - \frac{1}{y} \]

From the second equation:

\[ y = k - \frac{1}{z} \]

From the third equation:

\[ z = k - \frac{1}{x} \]

### Step 2: Setting up the relationships

Now we can express \( y \) and \( z \) in terms of \( x \) and \( k \):

Substituting \( z \) from the third equation into the second:

\[ y = k - \frac{1}{k - \frac{1}{x}} \]

### Step 3: Simplifying the expression

Now, let's simplify \( y \):

\[ y = k - \frac{1}{k - \frac{1}{x}} = k - \frac{x}{kx - 1} \]

### Step 4: Finding a common relationship

Now we can express \( x, y, z \) in a cyclic manner. We can also derive similar expressions for \( z \) and \( x \) using the same method.

### Step 5: Forming the ratios

From the equations, we can form the ratios:

1. From \( x - y = \frac{1}{z} - \frac{1}{y} \)
2. From \( y - z = \frac{1}{x} - \frac{1}{z} \)
3. From \( z - x = \frac{1}{y} - \frac{1}{x} \)

### Step 6: Multiplying the equations

Now we multiply all three equations:

\[ \frac{x - y}{y - z} \cdot \frac{y - z}{z - x} \cdot \frac{z - x}{x - y} = \frac{1}{xyz} \]

### Step 7: Cancelling terms

Notice that the left-hand side simplifies to 1:

\[ 1 = \frac{1}{xyz} \]

### Step 8: Solving for \( xyz \)

Taking the reciprocal gives us:

\[ xyz = 1 \]

### Final Answer

Thus, the value of \( xyz \) is:

\[ \boxed{1} \]