Question

If $\left(1\right)\left(2003\right)+\left(2\right)\left(2002\right)+\left(3\right)\left(2001\right)+\cdots +\left(2003\right)\left(1\right)=\left(2003\right)\left(334\right)\left(x\right),$ then the value of $\left[\frac{x}{5}\right]$ is equal to
([.] represents the greatest integer function)

Answer 1

Let $S=\left(1\right)\left(2003\right)+\left(2\right)\left(2002\right)+\left(3\right)\left(2001\right)+\cdots +\left(2003\right)\left(1\right)$
$K=1^2+2^2+3^2+\cdots +(2003{)}^2\Rightarrow S+K=1(2004)+2(2004)+3(2004)+\cdots +2003(2004)\Rightarrow S+K=(2004\left)\right(1+2+3+\cdots +2003)\Rightarrow S+K=2004\times \frac{2003\times 2004}{2}\Rightarrow (2003\left)\right(334\left)\right(x)+\frac{2003\times 2004\times 4007}{6}=2004\times \frac{2003\times 2004}{2}\Rightarrow (334\left)\right(x)+(334\left)\right(4007)=(2004\left)\right(1002)\Rightarrow (334\left)\right(x+4007)=(2004\left)\right(1002)\Rightarrow x+4007=6012\Rightarrow \frac{x}{5}=\frac{2005}{5}\therefore \left[\frac{x}{5}\right]=401$