Conditional Expectations Given Sum of I.I.D.

Hint: Since the random variables are exchangeable, all $E[X_i | \sum_{k=1}^n X_k]$ are equal. What is their sum?

For the late comers for this question. I think clear solution of the problem might be as follows:

Let $Y = \sum_{i=1}^n X_i$, by using the property of conditional expectation $\mathbb{E}[g(Y)|Y] = g(Y)$ (i.e.$\mathbb{E}[Y|Y] = Y$), we can write

$$ Y = \mathbb{E}[X_1 + X_2 + \cdots + X_n |Y] = \mathbb{E}[X_1|Y]+ \mathbb{E}[X_2|Y]+ \cdots + \mathbb{E}[X_n|Y] = n\mathbb{E}[X_1|Y],$$ since $X_i$'s are i.i.d., then $$\mathbb{E}[X_1|Y] = \frac{Y}{n}.$$