Source: 2017 Canadian Open Math Challenge, Problem A1
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The average of the numbers $2$, $5$, $x$, $14$, $15$ is $x$. Determine the value of $x$ .
1. We start with the given average formula for the numbers \(2\), \(5\), \(x\), \(14\), and \(15\). The average is given to be \(x\). Therefore, we can write the equation for the average as:
   \[
   \frac{2 + 5 + x + 14 + 15}{5} = x
   \]

2. Simplify the numerator on the left-hand side:
   \[
   \frac{2 + 5 + 14 + 15 + x}{5} = x
   \]
   \[
   \frac{36 + x}{5} = x
   \]

3. To eliminate the fraction, multiply both sides of the equation by 5:
   \[
   36 + x = 5x
   \]

4. Isolate \(x\) by subtracting \(x\) from both sides:
   \[
   36 = 5x - x
   \]
   \[
   36 = 4x
   \]

5. Solve for \(x\) by dividing both sides by 4:
   \[
   x = \frac{36}{4}
   \]
   \[
   x = 9
   \]

The final answer is \(\boxed{9}\).