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Question Number 215229 by Mingma last updated on 01/Jan/25

	Answered by devdutt last updated on 01/Jan/25
	$1. f((k/n)) = (n^(2025) /(n^(2025) +k^(2025) )) 2. f((k/n))+f((n/k)) = 1, f((n/n)) = f(1)= (1/2) 3. {(k/n)}_(k,n∈[45]) ⊃ {(k/n),(n/k)}, ∀k,n∈[45] 4. 2Σ_(n∈[45]) Σ_(k∈[45]) f((k/n)) = 2Σ_(n∈[45]) f((n/n)) + 2Σ_(n∈[45]) Σ_(k∈[45]−{n}) ((f(k/n)+f(n/k))/2) = 2×((45)/2) + 2×((45×44)/2) = 45^2 = 2025$
	$1 . f (\frac{k}{n}) = \frac{n^{2025}}{n^{2025} + k^{2025}}$ $2 . f (\frac{k}{n}) + f (\frac{n}{k}) = 1,$ $f (\frac{n}{n}) = f (1) = \frac{1}{2}$ $3 . {\frac{k}{n}}_{k, n \in [45]} \supset {\frac{k}{n}, \frac{n}{k}}, \forall k, n \in [45]$ $4 . 2 \sum_{n \in [45]} \sum_{k \in [45]} f (\frac{k}{n}) = 2 \sum_{n \in [45]} f (\frac{n}{n}) + 2 \sum_{n \in [45]} \sum_{k \in [45] - {n}} \frac{f (k / n) + f (n / k)}{2}$ $= 2 \times \frac{45}{2} + 2 \times \frac{45 \times 44}{2}$ $= 45^{2}$ $= 2025$
	Commented by Mingma last updated on 01/Jan/25
	Perfect ��
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