8. A coin collector has 100 coins that look the same. The collector knows that 30 of them are genuine, and 70 are fake, and that all genuine coins weigh the same, while all fake coins weigh different and are heavier than the genuine ones. The collector has a balance scale that can be used to compare the weight of two groups of coins of the same number. To ensure finding at least one genuine coin, how many weighings are needed at minimum?
8. First, it is clear that 70 weighings can certainly find at least one genuine coin.

In fact, each time, one coin is placed on each side of the balance. If they weigh the same, both are genuine; if they do not, the heavier one must be a counterfeit. Thus, each weighing either finds two genuine coins or one counterfeit.

After 70 weighings, either at least two genuine coins have been found, or all 70 counterfeits have been identified. This indicates that all genuine coins can be found.

Next, we prove that when the weight of the genuine coin is \(2^{100}\), and the weight of the counterfeit coins is
\[
m_{i}=2^{100}+2^{i} \quad (1 \leqslant i \leqslant 70),
\]
69 weighings cannot guarantee finding one genuine coin.
It is easy to see that during the weighing (let's say \(k\) coins on each side), the side with the heaviest counterfeit among the \(2k\) coins will be heavier.

Assume the collector follows his plan and is informed of the results of his weighings and the weights assigned to some counterfeit coins.

The first time he weighs, designate any one of the heavier side as weight \(m_{70}\).

Generally, when he weighs, if neither side has any counterfeit coins with already assigned weights, assign the heaviest unassigned weight to any one of the counterfeit coins on the heavier side; if there are counterfeit coins with previously assigned weights on the balance, do not assign any weight to any counterfeit coin this round.

After 69 rounds, the weight \(m_{1}\) has not been assigned, and the counterfeit coin with weight \(m_{1}\) is indistinguishable from 30 genuine coins.
In summary, finding one genuine coin requires at least 70 weighings.
(Li Weiguo provided)