- Find all real-valued functions f on the reals such that (1) f(1) = 1, (2) f(-1) = -1, (3) f(x) ≤ f(0) for 0 < x < 1, (4) f(x + y) ≥ f(x) + f(y) for all x, y, (5) f(x + y) ≤ f(x) + f(y) + 1 for all x, y.
- Given a nondegenerate triangle ABC, with circumcentre O, orthocentre H, and circumradius R, prove that |OH|< 3R.
- Let n be an integer of the form a2 +b2, where a and b are relatively prime integers and such
that if p is a prime, p ≤ √n, then p divides ab. Determine all such n.
- Is there an infinite set of points in the plane such that no three points are collinear, and the
distance between any two points is rational?
- You are given three lists A, B, and C. List A contains the numbers of the form 10k in base
10, with k any integer greater than or equal to 1. Lists B and C contain the same numbers
translated into base 2 and 5 respectively:
A B C
10 1010 20
100 1100100 400
1000 1111101000 13000
Prove that for every integer n > 1, there is exactly one number in exactly one of the lists Bor C that has exactly n digits.