Question

$2014 ^{x-2}+2014^{y+2} \leq 2*2014^{ \frac{x+y}{2}$
S=$2^x+2^y$
S este divizibil cu.....?

Answer 1

folosim Ma≥Mg
$\frac{ 2014^{x-2} + 2014^{y+2} }{2} \geq \sqrt{ 2014^{x-2}* 2014^{y+2} } =>$
$2014^{x-2} + 2014^{y+2} \geq 2* 2014^{ \frac{x-2+y+2}{2} }$
Avem 
$2014^{x-2} + 2014^{y+2} \leq 2* 2014^{ \frac{x+y}{2} }$   si
  $2014^{x-2} + 2014^{y+2} \geq 2* 2014^{ \frac{x+y}{2} }$
Din astea rezulta ca x=1 si y=1
$S= 2^{2} + 2^{2} =4+4=8$
S divizibila ca 2,4