Define the symbol # by the following equations:
`{:(x#y = (x - y)^(2),", if" x > y),(x#y = x + y//4,", if" x le y):}`
If `x # y = -1`, which of the following could be true?

Answer

To solve the problem, we need to analyze the defined function \( x \# y \) based on the given conditions. The function is defined as follows:

1. \( x \# y = (x - y)^2 \) if \( x > y \)
2. \( x \# y = \frac{x + y}{4} \) if \( x \leq y \)

We are given that \( x \# y = -1 \) and need to determine which conditions could be true.

### Step 1: Analyze the first case \( x > y \)

If \( x > y \), then:

\[ x \# y = (x - y)^2 \]

Since the square of any real number is non-negative, \( (x - y)^2 \geq 0 \). Therefore, it cannot equal -1.

**Conclusion for this case:** It is impossible for \( x \# y = -1 \) when \( x > y \).

### Step 2: Analyze the second case \( x \leq y \)

If \( x \leq y \), then:

\[ x \# y = \frac{x + y}{4} \]

We set this equal to -1:

\[ \frac{x + y}{4} = -1 \]

To solve for \( x + y \):

\[ x + y = -4 \]

This equation can hold true for various values of \( x \) and \( y \) as long as their sum is -4.

**Conclusion for this case:** It is possible for \( x \# y = -1 \) when \( x \leq y \).

### Step 3: Determine possible values for \( x \) and \( y \)

1. **Case when \( x = y \)**:
- If \( x = y \), then \( x + y = 2x = -4 \) implies \( x = -2 \) and \( y = -2 \).
- This satisfies \( x \leq y \).

2. **Case when \( x < y \)**:
- Choose \( x = -3 \) and \( y = -1 \):
- Then \( x + y = -3 + (-1) = -4 \), which also satisfies \( x \leq y \).

### Conclusion

From the analysis, we conclude that the only valid conditions for \( x \# y = -1 \) are:

- \( x = y \)
- \( x < y \)

Thus, the possible true statements are:

- \( x = y \)
- \( x < y \)