Solve this hard Romanian Mathematical Olympiad problem using Chebyshev Polynomials!

In this video we attempt a Romanian Mathematical olympiad problem. It requires proving a sum of binomial terms is always divisible by 2^(n-1). We first rewrite this sum in terms of a real part. Then we use demoirves theorem to simplify this real part. After this we spot induction should be possible. To create the induction step we use the Chebyshev Polynomial's recurrence relation. Everything crumbles after this point and the result is shown.