Question

The sum of the roots (real or complex) of the equation $x^{2001}+{\left(\frac{1}{2}-x\right)}^{2001}=0$ is

Answer 1

Answer: 500

$x^{2001}-\left[{}^{2001}{C}_0{x}^{2001}-{}^{2001}{C}_1{x}^{2000}\cdot \frac{1}{2}+{}^{2001}{C}_2{x}^{1999}\cdot \frac{1}{4}\dots \right]=0$
${}^{2001}{C}_1{x}^{2000}\cdot \frac{1}{2}-{}^{2001}{C}_2{x}^{1999}\cdot \frac{1}{4}\dots =0$
sum of the roots $={}^{2001}{C}_2\cdot \frac{1}{4}\cdot \frac{2}{{}^{2001}{C}_1}$
$=\frac{2001\cdot 2000}{4\cdot 2001}=500$