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 number (1,2,3,4,5,6) on the top of each bag should correspond to the weight of the coins in that bag. You have scales without any weights and without a scale. What is the least number of attempts to weigh you should make to be sure that all tag numbers are correct?

Author: Ilya Boguslavsky. Recommended for 9th-10th graders.

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