Search problems

2 problems match “telescoping products”. Clear

Visitors search a sample. Sign in for free to search every problem. Sign in

1. Final number from repeated operation ●●●○○ 50 ptsThe 97 numbers $\frac{49}{1}, \frac{49}{2}, \frac{49}{3}, \ldots, \frac{49}{97}$ are written on a blackboard. Repeatedly pick two numbers $a, b$ on the board and replace them by $2ab - a - b + 1$ until only one number remains. Find the final number.
2. Least number of factors giving 51/2010 ●●●●○ 100 ptsFind the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots, s_n\}$ consisting of $n$ distinct positive integers such that $$\left(1-\frac{1}{s_1}\right)\left(1-\frac{1}{s_2}\right)\cdots\left(1-\frac{1}{s_n}\right)=\frac{51}{2010}.$$