KöMaL 2017 A703

Let $n\ge 2$ be an integer. We call an ordered $n$-tuple of integers primitive if the greatest common divisor of its components is $1$. Prove that for every finite set $H$ of primitive $n$-tuples, there exists a non-constant homogenous polynomial $f(x_1,x_2,\dots,x_n)$ with integer coefficients whose value is $1$ at every $n$-tuple in $H$.