Higher Secondary P2

Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$.
1. Given the function \( g \) from the set of ordered pairs of real numbers to the same set, we have the property:
   \[
   g(x, y) = -g(y, x) \quad \text{for all real numbers } x \text{ and } y.
   \]
2. We need to find a real number \( r \) such that \( g(x, x) = r \) for all real numbers \( x \).

3. Substitute \( x = y \) into the given property:
   \[
   g(x, x) = -g(x, x).
   \]

4. This simplifies to:
   \[
   g(x, x) + g(x, x) = 0 \implies 2g(x, x) = 0 \implies g(x, x) = 0.
   \]

5. Therefore, the real number \( r \) such that \( g(x, x) = r \) for all real numbers \( x \) is:
   \[
   r = 0.
   \]

The final answer is \(\boxed{0}\).