Brazilian Mathematical Olympiad

We have $1^3 + 2^3 + \cdots + 10^3 = 3025$. Now $x_i$ is a positive integer solution to $(*)$ iff $y_i = x_i - 1$ are all non-negative and satisfy $(y_1+1)+2^3(y_2+1)+3^3(y_3+1)+\cdots+10^3(y_{10}+1) = 3025$ and hence $y_1+2^3y_2+3^3y_3+\cdots+10^3y_{10} = 0$. But that clearly has the unique solution $y_i = 0$, so the unique solution to $(*)$ is $x_i = 1$.