Question 33
Let $n \in \mathbb{N}$ be a positive integer. We call a function $f(x, y)$ a *friend* of $n$ if for at least one percent of positive integers $k$ such that $0 \le k \le n$ the equation $f(x, y) = k$ has a solution $(x_0, y_0)$ in positive integers such that $\frac{y_0}{x_0} \in [\frac{1}{100}, 100]$. Let $g(x, y)$ be a polynomial with non-negative real coefficients of total degree greater than $2$ such that $g(x, y) \le f(x, y)$, for all positive real numbers $x, y$ satisfying $\frac{y}{x} \in [\frac{1}{100}, 100]$. Prove that $f(x, y)$ would not be a *friend* of $n$ for all sufficiently large $n$.