Problem 1. For any integer and two matrices with real entries that satisfy the equation
prove that . Does the same conclusion follow for matrices with complex entries?
Problem 2. For a positive integer , let be the number obtained by writing in binary and replacing every with and vice versa. For example, is in binary, so is in binary, therefore . Prove that
When does the equality hold?
Problem 3. Let , and for .
Determine whether or not is a rational number.
Problem 4. Determine whether or not there exist integers such that
Problem 5. Let , and let be points in the dimensional Euclidean space, not lying on the same hyperplane, and let be a point strictly inside the convex hull of . Prove that holds for at least pairs with .