Let's do some calculus! (20)

Calculus Level 4

\[\begin{align} \lim_{n \to \infty} & \bigg( \int_{0}^{a} {\dfrac{f(x)}{f(x) + f(a-x)}} \ dx + a \int_{a}^{2a} {\dfrac{f(x)}{f(x) + f(3a-x)}} \ dx \\ & \quad + a^{2} \int_{2a}^{3a} {\dfrac{f(x)}{f(x) + f(5a-x)}} \ dx + \cdots \\ & \quad + a^{n-1} \int_{(n-1)a}^{na} {\dfrac{f(x)}{f(x) + f \big( \left(2n-1\right)a-x \big)}} \ dx \bigg) \\ & = \dfrac{7}{5} \end{align} \]

Given that function \(f(x) > 0\), \(\forall ~ x \in \mathbb{R}\) is bounded and satisfies the condition above. If \(a < 1\), find \(a\).


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