SOAL IMO PHILIPINA 2011

1. Find all nonempty finite sets X of real numbers with the following property:

x + ⎢x⎢ ∈ X  for all x ∈ X.

2. In ΔABC, let X and Y be the midpoints of AB and AC, respectively. On segment BC, there is a point D, different from its midpoint, such that ∠XDY = ∠BAC. Prove that AD is perpendicular to BC.

3. The 2011th prime number is 17483, and the next prime is 17489.

Does there exist a sequence of 2011²⁰¹¹ consecutive positive integers that contains exactly 2011 prime numbers? Prove your answer.

4. Find all (if there is one) functions f : ℝ ⟹ ℝ that satisfy the following

functional equation:

f(f(x)) + xf(x) = 1 for all x ∈ ℝ

5. The chromatic number of the (infinite) plane, denoted by χ, is the smallest number of colors with which we can color the points on the plane in such a way that no two points of the same color are one unit apart.