Use \to for a simple right arrow in LaTeX. Also, the last instance of $ f$ in your problem should of course be $ d$. [mod: this has been edited] Presumably the sphere and circle are meant to have the same radius? If so, an equilateral triangle inscribed in a great circle (= equator) of the sphere must be sent to an equilateral triangle inscribed in the circle. Now choose the north pole relative to the selected equator and it's clear there is nowhere for it to go. Or, if you don't want me to use points on the boundary, just begin with a slightly smaller sphere inside our larger one. If the circle can have arbitrarily larger radius, it's not clear what the answer should be (but in this case the problem probably belongs to the Calculus and Analysis forum, not here).

I think it doesn't belong to analysis forum and is a fine Olympiad problem. It can be solved (for any radius of the circle) without ideas of continuity, homeomorphism etc. Consider a cube of side length $a$ in the interior of the sphere. Clearly, all points from the cube are mapped to points on the given circle (say with diameter $D$). Now we set our coordinate system such that the origin is at a corner of the cube and the positive $x,y,z$ axes are along three adjacent sides of the cube. For a positive integer $n$, consider the set \[S_n=\{\left(\frac{ax}{n},\frac{ay}{n},\frac{az}{n}\right)|\ 0\le x,y,z\le n\}\]This set is a subset of the said cube. Let $f(S_n)$ denote the set of images of points in $S_n$ under $f$. Each point in $S_n$ is at least at a distance $d=a/n$ from all other points in $S_n$. So same must hold for $f(S_n)$. Now we consider disks of diameter $d$ centered at the points in $f(S_n)$. Clearly they are disjoint to one another. And they are all contained within a circle of diameter $D+d$ which is concentric with the given circle of diameter $D$. So the total area of those $(n+1)^3$ disks should be less than the area of that circle. i.e.\[(n+1)^3\frac{\pi}{4} \left(\frac{a}{n}\right)^2\le \frac{\pi}{4} (D+\frac{a}{n})^2\]\[\text{But we have }na^2<(n+1)^3\left(\frac{a}{n}\right)^2\le (D+\frac{a}{n})^2\le (D+a)^2\implies n<\left(1+\frac{D}{a}\right)^2\]But this must be true for all $n$, which is impossible. Hence no such function exists.

Very nice.