MathDB

Prove that there exist two sets A, B [Iran Second Round 95]

Source:

November 25, 2010

geometry3D geometryspherevectornumber theory proposednumber theory

Problem Statement

Prove that for every positive integer

n \geq 3

there exist two sets

A =\{ x_1, x_2,\ldots, x_n\}

and

B =\{ y_1, y_2,\ldots, y_n\}

for which
i)

A \cap B = \varnothing.

ii)

x_1+ x_2+\cdots+ x_n= y_1+ y_2+\cdots+ y_n.

ii)

x_1^2+ x_2^2+\cdots+ x_n^2= y_1^2+ y_2^2+\cdots+ y_n^2.