Suppose that we have two simple closed curves in $R^3$ which are linked.
And suppose that the distance between these curves is $1$.
Prove that the length of each curve is at least $2\pi$.

This problem has an interesting history. It was published in the book by W. Hayman, Unsolved problems
in Function theory, where it was attributed to F. Gehring. I solved it in 1977, jointly with Oleg Vinkovski, prepared a paper
and gave a seminar talk. After the talk, I was approached by an undergraduate student,
who proposed a ridiculously simple solution. Just two lines, using nothing.
So I did not submit my paper. Later I've seen several published solutions, but none of them
was so simple.

EDIT. Here is this proof (due to Igor Syutrik).
Fix a point $M$ on $A$.
Then one can find another point $M'$
on $A$ such that the interval $[M,M']$ intersects $B$.
Indeed, otherwise we can deform $A$ to $M$ moving straight
along these intervals $[M,M']$ and deformation will not
cross $B$. Let $O$ be a point on $[M,M']$ that belongs
to $B$. Let $A'$ be the central projection of $A$ from
$O$ onto the unit sphere around $O$. Then $A'$ passes
through two diametrically opposite points of the sphere
and thus its length is at least $2\pi$.

EDIT2. Exactly the same proof is published in the paper
Criticality for the Gehring link problem, Geometry & Topology 10 (2006) 2055–2115, where it is credited to Marvin Ortel.

EDIT3. Our original solution with Vinkovski also has been rediscovered since then. It can be seen in this file: http://www.math.purdue.edu/~eremenko/dvi/gehring.pdf
Thanks to Anton Petrunin for finding this file on my computer:-)