MathDB

1

Part of 2012 Danube Mathematical Competition

Problems(2)

ab+1,bc+1, cd+1, da+1 simultaneously perfect square

Source: Danube 2012 junior p1

7/22/2019
a) Exist a,b,c,∈Na, b, c, \in N, such that the numbers ab+1,bc+1ab+1,bc+1 and ca+1ca+1 are simultaneously even perfect squares ? b) Show that there is an infinity of natural numbers (distinct two by two) a,b,ca, b, c and dd, so that the numbers ab+1,bc+1,cd+1ab+1,bc+1, cd+1 and da+1da+1 are simultaneously perfect squares.
Perfect SquaresPerfect Squarenumber theory
max no of lattice points that covers a square of side n +1/(2n+1)

Source: Danube 2012 p1

7/22/2019
Given a positive integer nn, determine the maximum number of lattice points in the plane a square of side length n+12n+1n +\frac{1}{2n+1} may cover.
lattice pointssquarecoveringcombinatorial geometrycombinatorics