Time allowed: 4 hours.
No calculators to be used.
Each question is worth 7 points.
1. Let be the set of all -tuples where each is a subset of . Let denote the number of elements of the set A.
Find the number
2. Show that for any positive integers and , cannot be a power of .
3. Let be positive real numbers. Prove that
4. Let be a triangle and the foot of the altitude from . Let and be on a line through such that is perpendicular to is perpendicular to , and and are different from . Let and be the midpoints of the line segments and , respectively. Prove that is perpendicular to .
5. Determine the largest of all integers with the property that is divisible by all positive integers that are less than .