Problems(3)
2019 Saint Peteresburg Grade 11 P2
Source:
4/14/2019
On the blackboard there are written different positive integers . To each of these numbers is added the of the other numbers . In the new numbers , is it possible for of them to be equal. (С. Берлов)
combinatoricsnumber theorygreatest common divisor
k flights in n cities from 2 airlines
Source: St Petersburg 2019 10.2
5/1/2019
Every two of the cities of Ruritania are connected by a direct flight of one from two airlines. Promonopoly Committee wants at least flights performed by one company. To do this, he can at least every day to choose any three cities and change the ownership of the three flights connecting these cities each other (that is, to take each of these flights from a company that performs it, and pass the other). What is the largest committee knowingly will be able to achieve its goal in no time, no matter how the flights are distributed hour?
combinatoricsmaximum
2019 metro stations, create k more lines
Source: St. Petersburg 2019 9.2
5/2/2019
In the city built are metro stations. Some pairs of stations are connected. tunnels, and from any station through the tunnels you can reach any other. The mayor ordered to organize several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and in each station must lie at least on one line. To save money no more than lines should be made. It turned out that the order of the mayor is not feasible. What is the largest it could to happen?
combinatorics