MathDB

1

Part of 2022 Kosovo National Mathematical Olympiad

Problems(4)

22 coins

Source: Kosovo Mathematical Olympiad 2022, Grade 9, Problem 1

3/6/2022
Ana has 2222 coins. She can take from her friends either 66 coins or 1818 coins, or she can give 1212 coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have.
combinatorics
4 weights on a scale

Source: Kosovo Mathematical Olympiad 2022, Grade 10, Problem 1

3/6/2022
Ana has a scale that shows which side weight more or if both side are equal. She has 44 weights which look the same but they weight 1001g,1002g,1004g1001g, 1002g, 1004g and 1005g1005g, respectively. Is it possible for Ana to find out the weight of each of them with only 44 measurements?
combinatorics
22 light bulbs

Source: Kosovo Mathematical Olympiad 2022, Grade 11, Problem 1

3/6/2022
2222 light bulbs are given. Each light bulb is connected to exactly one switch, but a switch can be connected to one or more light bulbs. Find the least number of switches we should have such that we can turn on whatever number of light bulbs.
combinatorics
a+bc=b+ca=c+ab

Source: Kosovo Mathematical Olympiad 2022, Grade 12, Problem 1

3/6/2022
Find all real numbers a,ba,b and cc such that a+bc=b+ca=c+aba+bc=b+ca=c+ab.
algebra