MathDB

1

Part of 1995 Iran MO (2nd round)

Problems(1)

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

Source:

11/25/2010
Prove that for every positive integer n≥3n \geq 3 there exist two sets A={x1,x2,…,xn}A =\{ x_1, x_2,\ldots, x_n\} and B={y1,y2,…,yn}B =\{ y_1, y_2,\ldots, y_n\} for which
i) A∩B=∅.A \cap B = \varnothing.
ii) x1+x2+⋯+xn=y1+y2+⋯+yn.x_1+ x_2+\cdots+ x_n= y_1+ y_2+\cdots+ y_n.
ii) x12+x22+⋯+xn2=y12+y22+⋯+yn2.x_1^2+ x_2^2+\cdots+ x_n^2= y_1^2+ y_2^2+\cdots+ y_n^2.
geometry3D geometryspherevectornumber theory proposednumber theory