1

Part of 1988 Bundeswettbewerb Mathematik

Problems(2)

n^3 game pieces on n^4 fields of a square

Source: 1988 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

A square is divided into

n^4

fields like a chessboard.

n^3

game pieces are placed on these squares placed, on each at most one. There are the same number of stones in each row. Besides, the whole arrangement symmetrical to one of the diagonals of the square; this diagonal is called

d

. Prove that: a) If

n

is odd, then there is at least one stone on

d

n

is even, then there is an arrangement of the type described, in which there is no stone on

d

x-y,2x + 2y + 1,3x + 3y + 1 are pefrect squares if 2x^2 + x = 3y^2 + y

Source: 1988 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

For the natural numbers

x

y

2x^2 + x = 3y^2 + y

. Prove that then

x-y

2x + 2y + 1

3x + 3y + 1

are perfect squares.

number theoryPerfect SquaresPerfect Square