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Part of 2011 Baltic Way

Problems(1)

Prove that all x_i are equal

Source: Baltic Way 2011

11/6/2011
The real numbers x1,…,x2011x_1,\ldots ,x_{2011} satisfy x1+x2=2x1′, x2+x3=2x2′, …, x2011+x1=2x2011′x_1+x_2=2x_1',\ x_2+x_3=2x_2', \ \ldots, \ x_{2011}+x_1=2x_{2011}' where x1′,x2′,…,x2011′x_1',x_2',\ldots,x_{2011}' is a permutation of x1,x2,…,x2011x_1,x_2,\ldots,x_{2011}. Prove that x1=x2=…=x2011x_1=x_2=\ldots =x_{2011} .
algebra proposedalgebra