Function Sums to One

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A function \(f:\{1, 2, \cdots , 2014\} \rightarrow \mathbb{R^+} \) satisfies the following relation.

\[\sum \limits_{i=1; \ i \neq j}^{2014} f(i) f(j) = 1 \quad \forall \ j \in \{1, 2, \cdots , 2014 \} \] The sum of all possible values of \(f(2014)\) is \(k\). Find \( \dfrac{1}{k^2}\).

Details and assumptions

  • The range of \(f\) is \(\mathbb{R^+},\) i.e. \(f(x)\) is always positive.

  • If no such functions exist, \(k=0\).

  • If \(f(2014)\) can take only one value, \(k\) equals the only possible value of \(f(2014)\).

This problem appeared in the Proofathon Algebra contest, and was posed by Paramjit Singh.

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