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Number theory Difficulty 7.0 National Olympiad, round 2 Prove it Hungary

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 $H$ .

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