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2 problems match “partial sums”. Clear

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1. Ratio of geometric partial sums ●●○○○ 20 ptsLet $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$. If $a_{5}+2a_{10}=0$, find the value of $\frac{S_{20}}{S_{10}}$. Give your answer as a decimal.
2. Denominators of partial sums ●●●●○ 100 ptsOlya wrote fractions of the form $\frac{1}{n}$ on cards, where $n$ runs over all positive divisors of $6^{100}$ (including $1$ and $6^{100}$). She arranged the cards in some order. She then wrote on the board the number on the first card, the sum of the numbers on the first two cards, the sum of the numbers on the first three cards, and so on, up to the sum of the numbers on all the cards. Each sum was written as an irreducible fraction. What is the smallest possible number of different denominators that could appear on the board?