Solve the equation [x]{x}=x, where [] and {} denote the greatest integer function and fractional part, respectively.

Answer

To solve the equation \([x]\{x\} = x\), where \([x]\) denotes the greatest integer function and \(\{x\}\) denotes the fractional part of \(x\), we can follow these steps:

### Step 1: Understand the Functions
Recall that:
- The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- The fractional part function \(\{x\} = x - [x]\).

### Step 2: Rewrite the Equation
We start with the equation:
\[ [x]\{x\} = x \]
Substituting \(\{x\}\) with \(x - [x]\):
\[ [x](x - [x]) = x \]

### Step 3: Expand the Equation
Expanding the left side:
\[ [x]x - [x]^2 = x \]

### Step 4: Rearrange the Equation
Rearranging gives us:
\[ [x]x - [x]^2 - x = 0 \]
This can be rewritten as:
\[ [x]x - x = [x]^2 \]
Factoring out \(x\) on the left side:
\[ x([x] - 1) = [x]^2 \]

### Step 5: Solve for \(x\)
Now, we can express \(x\) as:
\[ x = \frac{[x]^2}{[x] - 1} \]
This gives us a formula for \(x\) in terms of \([x]\).

### Step 6: Analyze the Values of \([x]\)
Let \([x] = n\), where \(n\) is an integer. Then:
\[ x = \frac{n^2}{n - 1} \]
This is valid for \(n \neq 1\) (since division by zero is not allowed).

### Step 7: Determine the Range of \(x\)
Since \([x] = n\), we have:
\[ n \leq x < n + 1 \]
Substituting \(x\):
\[ n \leq \frac{n^2}{n - 1} < n + 1 \]

### Step 8: Solve the Inequalities
1. **Left Inequality**:
\[ n \leq \frac{n^2}{n - 1} \]
Multiplying both sides by \(n - 1\) (valid for \(n > 1\)):
\[ n(n - 1) \leq n^2 \implies n^2 - n \leq n^2 \implies -n \leq 0 \quad \text{(always true for } n > 0\text{)} \]

2. **Right Inequality**:
\[ \frac{n^2}{n - 1} < n + 1 \]
Multiplying both sides by \(n - 1\):
\[ n^2 < (n + 1)(n - 1) \implies n^2 < n^2 - 1 \implies 0 < -1 \quad \text{(not true)} \]
This means we need to check specific values of \(n\).

### Step 9: Check Integer Values
- For \(n = 2\):
\[ x = \frac{2^2}{2 - 1} = 4 \quad (valid, since \(2 \leq 4 < 3\) is false) \]

- For \(n = 3\):
\[ x = \frac{3^2}{3 - 1} = \frac{9}{2} = 4.5 \quad (valid, since \(3 \leq 4.5 < 4\) is false) \]

- For \(n = 4\):
\[ x = \frac{4^2}{4 - 1} = \frac{16}{3} \approx 5.33 \quad (valid, since \(4 \leq 5.33 < 5\) is false) \]

### Conclusion
The only valid integer solutions for \(n\) yield no valid \(x\) that satisfies the original equation. Therefore, the solution set is empty.