Question 149
Several positive integers are given, not necessarily all different. Their sum is $2003$. Suppose that $n_{1}$ of the given numbers are equal to $1$, $n_{2}$ of them are equal to $2$, $\ldots$, $n_{2003}$ of them are equal to $2003$. Find the largest possible value of
$$
n_{2}+2 n_{3}+3 n_{4}+\cdots+2002 n_{2003}
$$
$$
n_{2}+2 n_{3}+3 n_{4}+\cdots+2002 n_{2003}
$$