MathDB

2

Part of 2021 Durer Math Competition Finals

Problems(3)

least sum of properties of all cities

Source: 2021 Dürer Math Competition Finals Day 1 E2

1/2/2022
In the country of Óxisz the minister of finance observed at the end of the tax census that the sum of properties of any two neighboring city counted in dinar is divisible by 10001000, and she also observed that the sum of properties of all cities is also divisible by 10001000. What is the least sum of properties of all cities if the map of the cities looks as follows? https://cdn.artofproblemsolving.com/attachments/0/5/274730ebfdd52d0c3642dfbd0596fe587eb211.png Remark: The cities may have non-integer properties, but it is also positive. On the map the points are the cities, and two cities are neighboring if there is a direct connection between them.
combinatorics
2^n+2^{n+3} is a perfect square

Source: 2021 Dürer Math Competition Finals Day2 E2 https://artofproblemsolving.com/community/c2749870_

1/8/2022
Find the number of integers nn between 11 and 20212021 such that 2n+2n+32^n+2^{n+3} is a perfect square.
number theoryPerfect Square
in a french village the number of inhabitants is a perfect square

Source: 2021 Dürer Math Competition Finals Day2 E+2 https://artofproblemsolving.com/community/c2749870_

1/9/2022
In a french village the number of inhabitants is a perfect square. If 100100 more people moved in, then the number of people would be 1 1 bigger than a perfect square. If again 100100 more people moved in, then the number of people would be a perfect square again. How many people lives in the village if their number is the least possible?
number theoryPerfect Square