I'm trying to find unknown PDF f(x;c) and function p(c) that solve
$$p(c)\frac{c}{2} = (1-p(c))\int_0^1 (1-\frac{c}{x+1}) f(x;c)dx \tag{1}$$
for all $c\in [0,1]$. I know that $c$ is continuous and strictly decreasing in $[0,1]$, with $p(0)=1$ and $p(1)=0$. I am also fairly certain that $p(c)=\lim_{n \to \infty} \frac{P_n(c)}{Q_n(c)}$ where $(P_n)_{n\geq 1}$ and $(Q_n)_{n\geq 1}$ are unknown sequences of n-degree polynomials.
Additionally, $(1)$ is actually a special case of
$$p(c)\frac{ct}{t+1} = (1-p(c))\int_{1-t}^1 (1-\frac{ct}{x+t}) f(x;c) dx$$
valid for $t\in [0,1]$, where $t=1$ in $(1)$. I presented $(1)$ first because my hunch was that it would be an easier solve, though it may be underspecified on its own.
I also know that $f(0;c) \to \infty$ as $c \to 1$, and $f(1;c) \to 2$ as $c \to 0$. That is, $f(x;c)$ collapses to a PMF at $c=1$.
The last thing I have is a (numerically generated) plot of $p(c)$. I can also generate plots of $f(x;c)$ for any given $c$. ($c=0.25,0.5,0.75$ attached).
This is an attempt at computing a Nash Equilibrium for a (dis)continuous game. Any help at all would be tremendously appreciated, thank you.