Xét $(1-1)^{2n+1}=C_{2n+1}^0 -C_{2n+1}^1+C_{2n+1}^2 -C_{2n+1}^3 +... -C_{2n+1}^{2n+1}=0$
$\Rightarrow C_{2n+1}^0 + C_{2n+1}^2 + ...+C_{2n+1}^{2n} = C_{2n+1}^1 + C_{2n+1}^3 +...+C_{2n+1}^{2n+1}$
$\Rightarrow 2\bigg [ C_{2n+1}^1 + C_{2n+1}^3 +...+C_{2n+1}^{2n+1}\bigg ]= \bigg [ C_{2n+1}^0 + C_{2n+1}^2 + ...+C_{2n+1}^{2n} \bigg ]+\bigg [C_{2n+1}^1 + C_{2n+1}^3 +...+C_{2n+1}^{2n+1} \bigg]$
$= C_{2n+1}^0 +C_{2n+1}^1+C_{2n+1}^2 +C_{2n+1}^3 +... +C_{2n+1}^{2n+1}=(1+1)^{2n+1}=2^{2n+1}$
$\Rightarrow C_{2n+1}^1 + C_{2n+1}^3 +...+C_{2n+1}^{2n+1}=2^{2n} =1024 =2^{10}$
$\Rightarrow n= 5 \Rightarrow n^2 =25$