MathDB

4

Part of 2024 Bundeswettbewerb Mathematik

Problems(2)

Tying unit cubes to a thread and building a cuboid

Source: Bundeswettbewerb Mathematik 2024, Round 1 - Problem 4

3/8/2024
For positive integers pp, qq and rr we are given p⋅q⋅rp \cdot q \cdot r unit cubes. We drill a hole along the space diagonal of each of these cubes and then tie them to a very thin thread of length p⋅q⋅r⋅3p \cdot q \cdot r \cdot \sqrt{3} like a string of pearls.
We now want to construct a cuboid of side lengths pp, qq and rr out of the cubes, without tearing the thread. a) For which numbers pp, qq and rr is this possible? b) For which numbers pp, qq and rr is this possible in a way such that both ends of the thread coincide?
combinatoricscombinatorics proposed
Welcome back to Sikinia! Have a good flight...

Source: Bundeswettbewerb Mathematik 2024, Round 2 - Problem 4

9/13/2024
In Sikinia, there are 20242024 cities. Between some of them there are flight connections, which can be used in either direction. No city has a direct flight to all 20232023 other cities. It is known, however, that there is a positive integer nn with the following property: For any nn cities in Sikinia, there is another city which is directly connected to all these cities. Determine the largest possible value of nn.
combinatoricscombinatorics proposedgraph theory