jmc

Calculate the remainders when $1^2$, $2^2$, ..., $10^2$ are divided by 11 and sum them to find that the remainder when $1^2+2^2+\cdots +10^2$ is divided by 11 is the same as that of $1+4+9+5+3+3+5+9+4+1=44$, which is $\boxed{0}$.