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, Research 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:-)
EDIT 4. Recently published updated version of Hayman's problem list,
W. Hayman and E. Lingham, Research problems in Function theory, Springer 2019, says that the article by R. Osserman, The isoperimetric inequality, Bull AMS 84 (1978), contains on p. 1226 a survey of published solutions of this problem. It does not contain a solution as simple as Syutrik's solution.