Many problems in recent years were proven using incredibly advanced methods, like Fermat's Last Theorem. I'm interested about solutions to any non-trivial results which are elementary (i.e. someone with only a solid grasp on mathematics taught in secondary school should be able to understand the proof, even if it takes some effort) which were proven in the last 70 years or so. Proofs by counterexample don't count.
I was interested in this because I discovered the Haruhi problem, which essentially proved that every superpermutation had a lower bound on its length of $n!+(n-1)!+(n-2)!+n-3$ using fairly elementary methods. That got me wondering about if there were similar results like this.
