ÖMO 2009 Problem 2

(i) For positive integers $a<b$ , let $M(a,b)=\frac{\Sigma^{b}_{k=a}\sqrt{k^2+3k+3}}{b-a+1}$ . Calculate $[M(a,b)]$ (ii) Calculate $N(a,b)=\frac{\Sigma^{b}_{k=a}[\sqrt{k^2+3k+3}]}{b-a+1}$ .