Function Sums to One

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.