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 $i=0,1,\dots$, $(a\oplus b{)}_i=(a_i+b_i)\mathrm{mod}2$, 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 $(x+x+y)\mathrm{mod}2=y\mathrm{mod}2=(y+x+x)\mathrm{mod}2$.

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 $\ast$ by the formula $f(x)\ast f(y)=f(x\oplus y)$. Following from the construction, both conditions (1) and (2) still hold.