Question

Daca $a^{2}$+$b^{2}$ - 2a-4b+5=0 Calculati $a^{2005} +(b-a) ^{2005}$

Answer 1

$a^2+b^2-2a-4b+5=0 \\ \\ (a^2-2a+1)+(b^2-4b+4)=0 \\ \\ (a-1)^2+(b-2)^2=0. \\ \\ Cum~(a-1)^2 \geq 0~si~(b-2) \geq 0,~rezulta~ca~(a-1)^2+(b-2)^2 \geq 0. \\ \\ Dar~(a-1)^2+(b-2)^2=0 \Rightarrow (a-1)^2=0~si~(b-2)^2=0. \\ \\ Rezulta~\boxed{a=1}~si~ \boxed{b=2}~.$

$a^{2005}+(b-a)^{2005}=1^{2005}+(0-1)^{2005}=1+(-1)^{2005}= \\ \\ =1-1=0.$