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Equation.

Source: Greece National Olympiad 2000 , tst , Problem 3.

November 19, 2005
algebra proposedalgebra

Problem Statement

Let c1,c2,…,cn,b1,b2,…,bnc_1,c_2,\ldots ,c_n,b_1,b_2,\ldots ,b_n (n≥2)(n\geq 2) be positive real numbers. Prove that the equation ∑i=1ncixi−bi=12∑i=1nxi \sum_{i=1}^nc_i\sqrt{x_i-b_i}=\frac{1}{2}\sum_{i=1}^nx_i has a unique solution (x1,…,xn)(x_1,\ldots ,x_n) if and only if ∑i=1nci2=∑i=1nbi\sum_{i=1}^nc_i^2=\sum_{i=1}^nb_i.
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