Beyond Zudilin's Conjectured $q$-analog of Schmidt's problem

Using the methodology of (rigorous) {\it experimental mathematics}, we give a simple and motivated solution to Zudilin's question concerning a $q$-analog of a problem posed by Asmus Schmidt about a certain binomial coefficients sum. Our method is based on two simple identities that can be automatically proved using the Zeilberger and $q$-Zeilberger algorithms. We further illustrate our method by proving two further binomial coefficients sums.