3. The Elector George has 100 coins, some of which are counterfeit (possibly all or none). George can show the expert from 10 to 20 coins, and the expert will tell him how many of them are counterfeit. The problem is that the only expert in the entire region is Baron Münchhausen, and he exaggerates: the result given by the baron is always greater than the true number by some fixed (and unknown to George) natural number. The baron is not bothered by the fact that he might say "thirteen," for example, if he was given only twelve coins. Can George reliably determine which coins are counterfeit by consulting the expert fewer than 120 times?
Solution. It will work. We will submit $X$ random coins for expertise, and then the same $X$ plus 1 more. If the baron says the same number both times, the added coin is genuine; otherwise, it is counterfeit. This way, we can divide the coins into groups of 10, and check each group for counterfeits in 11 queries: first, submit 10 other coins, and then add one coin from the group being checked. The total number of queries will be $11 \cdot 10 = 110 < 120$.

## Criteria.

Full solution - 7 points.

Solutions where only one coin is submitted for each query - 0 points.

Identifying the number by which the baron lies - 0 points.