Let $X_1,\ldots,X_n$ be iid, positive and square integrable random variables. Further let $S:=\sum_{i=1}^nX_i$. I am trying to compute $\mathbb E[X_1^2|S]$. What i got so far is $$\frac{S^2}n=\frac1n\mathbb E[S^2|S]=\mathbb E[X_1^2|S]+(n-1)\mathbb E[X_1X_2|S]$$
Is there any clever trick to get rid of $\mathbb E[X_1X_2|S]$?
EDIT: A similar version is $$Var(S|S)=\mathbb E(S^2|S)-(\mathbb E(S|S))^2=0$$ but also $$0=Var(S |S)=n\,Var(X_1|S)+n\,(n-1)\,Cov(X_1,X_2|S).$$
That implies $$Var(X_1|S)=-(n-1)\,Cov(X_1,X_2|S)$$