# Function Sums to One

Level pending

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 $$f(2014)$$.

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