2010
Problem - 280
Find the remainder when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999}}$ is divided by $1000$.
Answer
109
Note that $\underbrace{99\cdots 9}_{k}\equiv 999\equiv -1\pmod{1000}$ when $k\ge 3$. The original expression is congruent to $$9\times 99\times \underbrace{(-1)(-1)\cdots(-1)}_{997}\equiv 891\times (-1)\equiv \boxed{109}\pmod{1000}$$