2008 AMC 10B Problem 5

Before doing any algebra, look at the two inputs. They might be related in a simple way.

1. Try numbers first: with $x = 3$, $y = 1$, we get $(x - y)^2 = 2^2 = 4$ and $(y - x)^2 = (-2)^2 = 4$. The two inputs are equal.
2. This always happens. The numbers $x - y$ and $y - x$ are opposites (one is the negative of the other), and squaring a number and its opposite gives the same result.
3. So both inputs are the same number; call it $t$. The rule $a \$b = (a - b)^2$ gives
$$t \$t = (t - t)^2 = 0^2 = 0.$$

The answer is $\boxed{\textbf{(A)}\ 0}$.

A made-up operation is just a formula: put the inputs in and simplify. Here the key is noticing that the two inputs are equal, so their difference is $0$ before you expand anything. When an expression contains both $x - y$ and $y - x$, remember that squaring makes them match.

Test the choices with numbers. With $x = 3$, $y = 1$, the answer is $4 \$4 = 0$. The choices give (A) $0$, (B) $9 + 1 = 10$, (C) $18$, (D) $2$, (E) $12$. Only (A) matches.

Because $y - x = -(x - y)$ and squaring erases the sign, $(y-x)^2 = (x-y)^2$. So the two inputs to the operation are equal; call them both $t$. Then
$$t \$t = (t - t)^2 = 0^2 = 0.$$

The answer is $\boxed{\textbf{(A)}\ 0}$.