Team Selection Test for IMO

One trivial solution is the constant function $f\equiv 0$. Let $f$ be a nontrivial function for which the conditions (1) and (2) hold. We will show that there exists a natural number $c$ and a prime number $p$ for which it holds that $f(a)=c{v}_p(a)$ for each $a\in {\mathbb{Z}}^{\ast }$, where $v_p(a):=\text{the exponent of }p\text{ in the canonical factorization of }a$:

let us note at first that $f(1)=f(-1)=0$ (proof: $$f(1)=f(1\cdot 1)=f(1)+f(1),f(1)=f((-1)\cdot (-1))=f(-1)+f(-1);$$ from this and from (2) it follows that there exists a prime number $p$ for which $f(p)\ne 0$; for $c:=f(p)$ we will show that $f(a)=c{v}_p(a)$ holds for every $a\in {\mathbb{Z}}^{\ast }$; namely, for each prime $q\ne p$ there exists nonzero integers $a,\beta$ for which $1=ap+\beta q$, so that the inequality $0=f(ap+\beta q)\ge \min \{f(ap),f(\beta q)\}$ holds; from $$f(ap)=f(a)+f(p)\ge f(p)=c\ne 0$$ it follows that $f(\beta q)=0$ and $f(q)=0$; let $a=\pm p^k{q}^{\beta }{r}^{\gamma }...$ be the canonical factorization of $a$; then $$f(a)=f(\pm p^k)+f(q^{\beta })+f(r^{\gamma })+\cdots =f(\pm 1)+f(p^k)=kf(p)=c{v}_p(a)$$ It remains to note that each such function satisfies the conditions (1) and (2), and therefore it represents a nontrivial solution to the given problem.