MathDB

1

Part of 2020 Bundeswettbewerb Mathematik

Problems(2)

Putting quite a bit of gold in two treasure chests

Source: Bundeswettbewerb Mathematik 2020, Round 2 - Problem 1

11/17/2020
Leo and Smilla find 20202020 gold nuggets with masses 1,2,…,20201,2,\dots,2020 gram, which they distribute to a red and a blue treasure chest according to the following rule:
First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this chest.
This is repeated until all the nuggets are distributed. Finally, Smilla chooses one of the chests and wins all the nuggets from this chest.
How many gram of gold can Smilla make sure to win?
combinatoricscombinatorics proposedgameCombinatorial games
Perfect squares as sums or differences of powers

Source: Bundeswettbewerb Mathematik 2020, Round 1 - Problem 1

11/17/2020
Show that there are infinitely many perfect squares of the form 50m−50n50^m-50^n, but no perfect square of the form 2020m+2020n2020^m+2020^n, where mm and nn are positive integers.
number theorynumber theory proposed