IMO Team Selection Contest

Define an operation $\oplus$ on the set of all non-negative integers, which maps two non-negative integers $a$ and $b$ to a non-negative integer $a \oplus b$, such that for all $i = 0, 1, \dots$, $(a \oplus b)_i = (a_i + b_i) \bmod 2$, where $n_i$ stands for the binary digit corresponding to $2^i$ in the binary representation of $n$. This operation satisfies condition (1) for all non-negative integers because addition modulo 2 satisfies it. The operation also satisfies condition (2) because if $x, y \in \{0,1\}$, then $(x + x + y) \bmod 2 = y \bmod 2 = (y + x + x) \bmod 2$.

As the set of non-negative integers as well as the set of all integers are countable, there exists one-to-one correspondence $f$ between these sets (e.g. mapping a non-negative integer $x$ to the integer $(-1)^x \lfloor \frac{x+1}{2} \rfloor$). Every integer can therefore be uniquely expressed in the form $f(n)$, where $n$ is a non-negative integer. Therefore we can define the operation $*$ by the formula $f(x) * f(y) = f(x \oplus y)$. Following from the construction, both conditions (1) and (2) still hold.