MathDB

1

Part of 2024 Bundeswettbewerb Mathematik

Problems(2)

Solving (x+2)^4-x^4=y^3 over the integers

Source: Bundeswettbewerb Mathematik 2024, Round 2 - Problem 1

9/13/2024
Determine all pairs (x,y)(x,y) of integers satisfying (x+2)4−x4=y3.(x+2)^4-x^4=y^3.
number theorynumber theory proposedDiophantine equation
Two-player game on a 7x7 board

Source: Bundeswettbewerb Mathematik 2024, Round 1 - Problem 1

3/8/2024
Arthur and Renate play a game on a 7×77 \times 7 board. Arthur has two red tiles, initially placed on the cells in the bottom left and the upper right corner. Renate has two black tiles, initially placed on the cells in the bottom right and the upper left corner. In a move, a player can choose one of his two tiles and move them to a horizontally or vertically adjacent cell. The players alternate, with Arthur beginning. Arthur wins when both of his tiles are in horizontally or vertically adjacent cells after some number of moves. Can Renate prevent him from winning?
combinatoricscombinatorics proposedgamewinning strategy