My favourite open problems

• For every polyhedron, does there exist a face-to-face partition into acute tetrahedra (in which all dihedral angles are acute)?
• Is there a face-to-face partition of R⁴ into acute simplices?
• What are all simplicial space-fillers in R^d?
• For every composite Fermat number F_m = 2^{2^m} + 1, does there exist a positive integer h such that F_m is divisible by 5h2^m+2 + 1?
• Given any odd integer k > 2 that is not a Sierpinski number, does there exist a Fermat number F_m having a prime factor of the form k2ⁿ + 1?
• Is a Fermat number F_m prime if F_m+1 is prime?