Let $f$ be a continuous function from $S^n$ to $R^n$. The Borsuk-Ulam theorem states that there is an $x$ such that $f(x)=f(-x)$.

Under what conditions is this true that for every constant ''r>0'' there is an $x$ such that $f(x)=r f(-x)$?

1. In general this is of course not true, for example when $f$ is a constant function.

2. What if $f$ is a non-negative function with at least one zero in $S^n$? As commented by @FrancescoPolizzi, this is insufficient, for example when $f(x)=\text{Re}(x)^2$, $f$ has two zeros, but for every $x$ on the unit sphere: $f(x)=f(-x)$.

3. What if there is at least one point $x_0$ for which $f(x_0)=0$ and $f(-x_0) > 0$ componentwise? I think for $n=1$ this is true. Choose a small $\epsilon>0$. Define $g(x)=f(x)/[\epsilon+f(-x)]$. Then $g(x_0)=0$ but $g(-x_0)=f(-x_0)/\epsilon$. This can be made arbitrarily large by a proper selection of $\epsilon$. The function $g$ is continuous, so by the Intermediate Value Theorem it must accept every value between 0 and $\infty$ somewhere between $x_0$ and $-x_0$.

Is this true in general? If not, what conditions should be added to make it true?