Ottó decided that for every pair of numbers $(x, y)$, he would assign a number denoted by $(x \circ y)$. He wants to achieve that the following relationships hold:
a) $x \circ y=y \circ x$
b) $(x \circ y) z=x z \circ y z$
c) $(x \circ y)+z=(x+z) \circ(y+z)$.

What should Ottó assign to the pair $(1975,1976)$?
Consider the number $(x \circ y)-x+(y \circ x)-y$. First, transform it using c), then b):

\[
\begin{gathered}
(x \circ y)-x+(y \circ x)-y=(x-x) \circ(y-x)+(y-y) \circ(x-y)= \\
=(-1) \cdot(0 \circ(x-y))+0 \circ(x-y)=0 .
\end{gathered}
\]

Finally, using a) from this, we get $x \circ y=(x+y) / 2$. Thus, Ottó's operation can only be the arithmetic mean of the two numbers. It is easy to verify that this indeed satisfies the properties a), b), and c). Therefore, the number assigned to (1975, 1976) must be 1975.5.

Mihály Fodor (Nyíregyháza, Zrínyi I. Gymnasium, 2nd grade)