IZhO 2005

For the positive real numbers $a,b,c$ prove that $$\frac c{a + 2b} + \frac d{b + 2c} + \frac a{c + 2d} + \frac b{d + 2a} \geq \frac 43.$$

The inner point $X$ of a quadrilateral is observable from the side $YZ$ if the perpendicular to the line $YZ$ meet it in the colosed interval $[YZ].$ The inner point of a quadrilateral is a $k-$ point if it is observable from the exactly $k$ sides of the quadrilateral. Prove that if a convex quadrilateral has a 1-point then it has a $k-$ point for each $k=2,3,4.$

Find all prime numbers $p,q < 2005$ such that $q | p^{2} + 8$ and $p|q^{2} + 8.$