MathDB

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Part of 2008 Poland - Second Round

Problems(2)

Consecutive integers of the form x^3+2y^2

Source: Polish MO 2008 Second Round (1st day)

2/22/2008
Determine the maximal possible length of the sequence of consecutive integers which are expressible in the form x^3\plus{}2y^2, with x,y x, y being integers.
number theory proposednumber theory
A board n on n

Source: Polish Mathematical Olympiad Second Round (day 2)

2/23/2008
We have an n×nn \times n board, and in every square there is an integer. The sum of all integers on the board is 00. We define an action on a square where the integer in the square is decreased by the number of neighbouring squares, and the number inside each of the neighbouring squares is increased by 1. Determine if there exists n≥2n\geq 2 such that we can turn all the integers into zeros in a finite number of actions.
geometryrectanglecombinatorics proposedcombinatorics