Subcontests
(3)box with 2015 white and 2015 black balls, choses two, remain 3 at the end
Vaggelis has a box that contains 2015 white and 2015 black balls. In every step, he follows the procedure below:
He choses randomly two balls from the box. If they are both blacks, he paints one white and he keeps it in the box, and throw the other one out of the box. If they are both white, he keeps one in the box and throws the other out. If they are one white and one black, he throws the white out, and keeps the black in the box.
He continues this procedure, until three balls remain in the box. He then looks inside and he sees that there are balls of both colors. How many white balls does he see then, and how many black? x^2y/z+y^2z/x+z^2x/y+2(y/xz+z/xy+x/yz)>= 9 if x,y,z>0
a) Prove that, for any real x>0, it is true that x3−3x≥−2 .
b) Prove that, for any real x,y,z>0, it is true that
zx2y+xy2z+yz2x+2(xzy+xyz+yzx)≥9 . When we have equality ?