MathDB

4

Part of 2022 239 Open Mathematical Olympiad

Problems(2)

Problem 4

Source: 239-School Open Olympiad (Senior Level)

4/25/2022
Vasya has a calculator that works with pairs of numbers. The calculator knows hoe to make a pair (x+y,x)(x+y,x) or a pair (2x+y+1,x+y+1)(2x+y+1,x+y+1) from a pair (x,y).(x,y). At the beginning, the pair (1,1)(1,1) is presented on the calculator. Prove that for any natural nn there is exactly one pair (n,k)(n,k) that can be obtained using a calculator.
number theoryprimesNatural Numbers
Nice graph theory

Source: 239-School Open Olympiad 2022, Junior League P4

4/19/2023
The degrees of all vertices of a graph are not less than 100 and not more than 200. Prove that its vertices can be divided into connected pairs and triples.
combinatoricsgraph theory