Let $N\in\{2,3,...\}$ and arbitrarily fix $(\alpha_i,\beta_i,\gamma_i,\delta_i)\in(0,1]^4$ for each $i\in\{1,...,N\}$. Then consider the following constrained optimization problem:
\begin{cases} \max\limits_{x_{1},...,x_N} & \sum_{i=1}^N \frac{1}{1+\frac{\alpha_i x_i + \gamma_i}{\beta_i x_i + \delta_i}}x_i\\ \ \hfill\ \ \ \ \text{s.t.} & x_i\in[0,1]\ \ \forall i\in\{1,...,N\}\, .\\ & \sum_{i=1}^Nx_i =1 \end{cases}
I am having a lot of difficulty in making a dent in solving this. I have tried mathematical induction, but to no avail. Trying to "directly" solve this also seems foolhardy. If anyone knows how to solve this, I would really appreciate the help!
Edit: I am interested analytically deriving the solution to the above optimization problem. (If a closed-form solution is possible, even better!) Many thanks to Max below for their clarifying question.
Only a partial answer, but too long for a comment. You can simplify your objective, since $$\frac{1}{1+\frac{\alpha_i x_i+\gamma_i}{\beta_i x_i + \delta_i}}x_i=\frac{\beta_i x+\delta_i}{(\alpha_i+\beta_i)x_i+(\gamma_i+\delta_i)}x_i=\frac{\beta_i}{\alpha_i+\beta_i}x_i+\frac{\delta_i-\frac{\beta_i(\gamma_i+\delta_i)}{\alpha_i+\beta_i}}{(\alpha_i+\beta_i)x_i+(\gamma_i+\delta_i)}x_i={\frac{\beta_i}{\alpha_i+\beta_i}x_i+\frac{\delta_i-\frac{\beta_i(\gamma_i+\delta_i)}{\alpha_i+\beta_i}}{\alpha_i+\beta_i}-\frac{(\gamma_i+\delta_i)\left(\delta_i-\frac{\beta_i}{\alpha_i+\beta_i}\right)}{(\alpha_i+\beta_i)((\alpha_i+\beta_i)x_i+(\gamma_i+\delta_i))}.}$$ That means you can write your objective in the form $\sum_{i=1}^N a_i x_i + \frac{1}{b_i x_i + c_i}$, where $b_i$ and $c_i$ have the same sign.