10. Let $x_{1}, x_{2}, \ldots, x_{1970}$ be positive integers satisfying $x_{1}+x_{2}+\cdots+x_{1970}=2007$. Determine the largest possible value of $x_{1}^{3}+x_{2}^{3}+\cdots+x_{1970}^{3}$.
10. Ans: 56841
We may assume $x_{1} \leq x_{2} \leq \cdots \leq x_{1970}$. For $1 \leq x_{i} \leq x_{j}$ with $i \leq j$, we have $x_{i}^{3}+x_{j}^{3} \leq x_{i}^{3}+x_{j}^{3}+3\left(x_{j}-x_{i}\right)\left(x_{j}+x_{i}-1\right)=\left(x_{i}-1\right)^{3}+\left(x_{j}+1\right)^{3}$. Thus when $x_{1}=x_{2}=\cdots=x_{1969}=1$ and $x_{1970}=38$, the expression $x_{1}^{3}+x_{2}^{3}+\cdots+x_{1970}^{3}$ attains its maximum value of $1969+38^{3}=56841$.