2 Olympiad Geometry problems I like

[sirio11]

Problem #1:

Let P_1,P_2,P_3,.......,P_n points contained in a circle with radius = r with P_1 being the center of the circumference.
Let d_k = min{//P_j - P_k// j <> k } (This is the minimum of the distances from the other points to P_k)

Prove that: (d_1)^2+(d_2)^2+(d_3)^2+.......+(d_n)^2 <= 9(r^2)

Problem # 2 (Hard):

Let abcdef an hexagon with every pair of sides parallel. Any 3 vertices can be covered by a wide stripe of length = 1 (This is the length between the lines of the infinite strip).

Prove that the hexagon can be covered by a wide stripe of length = sqrt(2)