We are given a number $n$ and a vector $p=(p_1,p_2,\ldots,p_r)$, where $$p_1\geq p_2 \geq \ldots \geq p_r \geq 0 ; \ \ \ \ \sum_{i\in [r]} p_i \leq 1$$

I'm interested in proving that a solution for the following set of equation exist (the variables here are $t_1,t_2,\ldots, t_r$): $$\begin{cases} p_i\cdot (1-(1-t_i)^n)\cdot t_{i+1}=p_{i+1}\cdot (1-(1-t_{i+1})^n)\cdot t_{i} & & & &\forall i\in [r-1] \\ \sum_{i\in [r]}t_i=1\end{cases}$$

So my questions are:

1. Is there always a solution $t$ for this set of equations?

2. If not, can we characterize when (i.e. for which inputs $n,p$) such solution exist?

3. (least important) In which cases such solution can be found efficiently (computationally)?

A few observations about this problem:

• If $n=2$, this is gives a set of $r$ linearly-independent linear equations, hence a solution exist and can be found efficiently.

• If $r=2$ (two variables) then this reduces to this question (for $p_2=1-p_1$), although now I'm also interested in a solution existence, which is true for $r=2$, but I'm not sure about general $r$. As a result of the discussion on the other question, I believe efficient computation for the general problem would be hard, but I'm wondering what about the existence of a solution.

• The first equation may be rewritten as $$p_i\cdot \sum_{k=1}^{n-1}(t_i)^k = p_{i+1}\cdot \sum_{k=1}^{n-1}(t_{i+1})^k $$