Question 104
Natural numbers $1, 2, 3, \ldots, 2003$ are written in an arbitrary sequence $a_{1}, a_{2}, a_{3}, \ldots, a_{2003}$. Let $b_{1} = 1 a_{1}$, $b_{2} = 2 a_{2}$, $b_{3} = 3 a_{3}$, $\ldots$, $b_{2003} = 2003 a_{2003}$, and $B$ be the maximum of the numbers $b_{1}, b_{2}, b_{3}, \ldots, b_{2003}$.
a) If $a_{1} = 2003, a_{2} = 2002, a_{3} = 2001, \ldots, a_{2002} = 2, a_{2003} = 1$, find the value of $B$.
b) Prove that $B \geq 1002^{2}$.
a) If $a_{1} = 2003, a_{2} = 2002, a_{3} = 2001, \ldots, a_{2002} = 2, a_{2003} = 1$, find the value of $B$.
b) Prove that $B \geq 1002^{2}$.