RO-NMO 2013

Solve the following equation ${{2}^{{{\sin }^{4}}x-{{\cos }^{2}}x}}-{{2}^{{{\cos }^{4}}x-{{\sin }^{2}}x}}=\cos 2x$

To be considered the following complex and distinct $a,b,c,d$ . Prove that the following affirmations are equivalent: i)For every $z\in \mathbb{C}$ the inequality takes place : $\left| z-a \right|+\left| z-b \right|\ge \left| z-c \right|+\left| z-d \right|$ ; ii)There is $t\in \left( 0,1 \right)$ so that $c=ta+\left( 1-t \right)b$ si $d=\left( 1-t \right)a+tb$

Find all injective functions $f:\mathbb{Z}\to \mathbb{Z}$ that satisfy: $\left| f\left( x \right)-f\left( y \right) \right|\le \left| x-y \right|$ ,for any $x,y\in \mathbb{Z}$ .