Question 76
Let $f:\mathbb R\to\mathbb R$ be an arbitrary function and $g:\mathbb R\to\mathbb R$ a quadratic function with the following property: for any real numbers $m,n$, the equation $f(x)=mx+n$ has solutions if and only if $g(x)=mx+n$ has solutions. Show that $f$ and $g$ are equal.
Second source statement (Hungarian version): Let $f:\mathbb R\to\mathbb R$ be arbitrary and $g:\mathbb R\to\mathbb R$ be a quadratic function such that for any real $m,n$, $f(x)=mx+n$ has a solution exactly when $g(x)=mx+n$ has a solution. Prove that the functions $f$ and $g$ are equal.
Second source statement (Hungarian version): Let $f:\mathbb R\to\mathbb R$ be arbitrary and $g:\mathbb R\to\mathbb R$ be a quadratic function such that for any real $m,n$, $f(x)=mx+n$ has a solution exactly when $g(x)=mx+n$ has a solution. Prove that the functions $f$ and $g$ are equal.