Examples of famous problems resolved easily Have there been examples of seemingly long standing hard problems, answered quite easily possibly with tools existing at the time the problems were made? More modern examples would be nice. An example could be Hilbert's basis theorem. Another could be Dwork's p-adic technique for rationality of zeta functions from algebraic varieties.

Finite field Kakeya "conjecture"
Let $\mathbb{F}_q$ be the finite field with $q$ elements and $A\subset \mathbb{F}_q^n$ be a set which contains a line in every possible direction.  Then there exists constant $c>0$ (depending only on $n$) such that
$$|A|\geq cq^n.$$
Dvir's proof of this is extremely elementary and easy, but nevertheless ingenious.  The whole paper is 6 pages long, but one could easily fit the whole argument in less than a page. See here.
