Problems for Math Olympiads

This problem was offered at the Brighton Beach (Brooklyn, New York) Mathematical Olympiad for 8th graders in 2000.

You have 6 bags with coins that look the same. Each bag contains sufficiently large number of coins. Coins from one bag weigh 1 gram each, coins from another bag weigh 2 grams each, ... coins from the last bag weigh 6 grams each. The tag (1,2,3,4,5,6) attached to each bag indicates the weight of the coins in that bag. You have scales without weights. What is the least number of weighings you need to confirm that all the tags have been attached correctly?

This problem was offered at the 1991 Moscow Mathematical Olympiad for 8th graders.

Same problem as above but each bag contains only one coin.

A more complicated version of the Ali-baba problem.

The Ali-baba problem is usually formulated with Ali-baba able to see the signs upon placing his hands on two sections of the table. What if the signs were not visible? Everything else remains the same: Ali-baba selects two sections, then by will can flip zero, one, or both signs, but when flipping a sign he no longer knows whether it is from a plus to a minus or the other way around. Is the goal still achievable? Both versions have been implemented interactively here.