Two problems concerning uniform polynomial approximation of continuous functions

We remind two theorems closely connected with the fundamental P. L. Chebyshev's theorem on the best approximation of functions by polynomials, namely S. N. Bernstein's theorem on reconstruction of a function by its deviations from polynomials, and the author's one on distribution of Chebyshev's alternance points. In connection with this two results two open (in author's opinion) problems are formulated.