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Find all the real numbers satisfying the equiation

Source: MTRP 2015 Class 11-Short Answer Type Question: Problem 4 :-

September 22, 2020
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Problem Statement

Find all real numbers x1,x2,⋯ ,xnx_{1}, x_{2}, \cdots, x_{n} satisfying,x1−12+2x2−22+⋯+nxn−n2=12(x1+x2+⋯+xn)\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\cdots+n \sqrt{x_{n}-n^{2}}=\frac{1}{2}\left(x_{1}+x_{2}+\cdots+x_{n}\right)
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