**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 .