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12 problems match “ratios”. Clear

Minimum distinct values in a ratio chain ●●●○○ 50 ptsLet $a_{1}, a_{2}, \ldots, a_{2006}$ be positive integers (not necessarily distinct) such that the 2005 ratios $\frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \ldots, \frac{a_{2005}}{a_{2006}}$ are all distinct. What is the least possible number of different values among $a_{1}, a_{2}, \ldots, a_{2006}$ ?
Minimum respondents on the twelfth day ●●●○○ 50 ptsDuring the first eleven days, 700 people responded to a survey question. Each person chose exactly one of three options, and the ratio of the numbers of responses was $4:7:14$ . On the twelfth day, several more people responded, so that the number of responses for each option did not decrease and the ratio became $6:9:16$ . What is the minimum number of people who must have responded on the twelfth day?