I was recently browsing and came upon this document which gives some open problems involving Diophantine Equations. Document: http://www.math.leidenuniv.nl/~evertse/07-workshop-problems.pdf
Upon searching a bit, I found that the first problem given has been solved. Is the second problem also solved? The problem is Find all $x,y \in \mathbf{N}$ such that $\binom{x}{2} = \binom{y}{5}$.
According to Blokhuis, Brouwer, and de Weger, Binomial collisions and near collisions, Integers 17 (2017) #A64, the question was settled in Bugeaud, Mignotte, Siksek, Stoll, and Tengely, Integral points on hyperelliptic curves, Algebra Number Theory 2 (2008) 859-885; there are no nontrivial solutions, other than those given in the comment.