RO-NMO 2005 Problem 84

a) Prove that for all positive reals $u,v,x,y$ the following inequality takes place: $$\frac ux + \frac vy \geq \frac {4(uy+vx)}{(x+y)^2} .$$ b) Let $a,b,c,d>0$ . Prove that $$\frac a{b+2c+d} + \frac b{c+2d+a} + \frac c{d+2a+b} + \frac d{a+2b+c} \geq 1.$$

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