Examples of open problems solved through short proof Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with some effort but went unnoticed in mathematics until someone provided a footnote with the 'obvious' solution? Proof needs to be reasonably short than that possibly expected.
Euler bridge problem qualifies however I am looking at something that would be published post 1900s or preferably 1950s in mathematics.
 A: The Stanley-Wilf Conjecture, that the number of permutations of $\{1,2, \ldots, n\}$ avoiding a fixed set of patterns grows at most exponentially with $n$, was open for fifteen or so years until a very short and elegant proof was found by Marcus and Tardos in 2004.
A: a semi famous example from number theory, Erdos Proof of Bertrands postulate (paper by Galvin). 

In 1845 Bertrand postulated that there is always a prime between n and 2n, and
  he verified this for $n < 3 \cdot 10^6$
  . Tchebychev gave an analytic proof of the postulate in
  1850. In 1932, in his first paper, Erdős gave a beautiful elementary proof using nothing
  more than a few easily verified facts about the middle binomial coefficient. We describe
  Erdős’s proof and make a few additional comments, including a discussion of how the
  two main lemmas used in the proof very quickly give an approximate prime number
  theorem. We also describe a result of Greenfield and Greenfield that links Bertrand’s
  postulate to the statement that $\{1, . . . , 2n\}$ can always be decomposed into $n$ pairs such
  that the sum of each pair is a prime.

A: I think Hilbert's Basis theorem is a good example. Mathematicians struggled with a more special question than that Hilbert proved. But there was a lot of critic of the proof, that was thought of as religion rather than mathematics.
A: Fundamental theorem of algebra. It was important enough for Gauss to do his thesis on it but an undergraduate course in complex analysis will teach you multiple very short proofs. Rouché's theorem is my personal favorite method.
A hugely important theorem can now be proved in a few lines (although I wouldn't call this an open problem).
A: Dvir's solution to Kakeya Conjecture over finite fields is probably a good example of this. See http://arxiv.org/abs/0803.2336 and https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/. 
