MathDB

3

Part of 2023 Durer Math Competition (First Round)

Problems(3)

12 Olympians and 88 humans in a Greek village

Source: (2022 -) 2023 XVI Dürer Math Competition Regional E2

5/25/2024
In a Greek village of 100100 inhabitants in the beginning there lived 1212 Olympians and 8888 humans, they were the first generation. The Olympians are 100%100\% gods while humans are 0%0\% gods. In each generation they formed 5050 couples and each couple had 22 children, who form the next generation (also consisting of 100100 members). From the second generation onwards, every person’s percentage of godness is the average of the percentages of his/her parents’ percentages. (For example the children of 25%25\% and 12.5%12.5\% gods are 18.75%18.75\% gods.) a) Which is the earliest generation in which it is possible that there are equally many 100%100\% gods as 0% 0\% gods? b) At most how many members of the fifth generation are at least 25% gods?
algebranumber theory
a positive integer next to points

Source: (2022 -) 2023 XVI Dürer Math Competition Regional E3

5/25/2024
Pythagoras drew some points in the plane and and connected some of these with segments. Now Tortillagoras wants to write a positive integer next to every point, such that one number divides another number if and only if these numbers are written next to points that Pythagoras has connected.Can Tortillagoras do this for the following drawings?
In part b), vertices in the same row or column but not adjacent are not connected. https://cdn.artofproblemsolving.com/attachments/1/e/7356e39e44e45e3263275292af3719595e2dd2.png
combinatorics
max different xy = prime. combo NT

Source: (2022 -) 2023 XVI Dürer Math Competition Regional E+3

5/25/2024
Let n≥3n \ge 3 be an integer and AA be a subset of the real numbers of size n. Denote by BB the set of real numbers that are of the form x⋅y x \cdot y, where x,y∈Ax, y \in A and x≠yx\ne y. At most how many distinct positive primes could BB contain (depending on nn)?
combinatoricsnumber theory