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 $t_1,t_2,\ldots, t_r$):
$$\begin{cases} p_i\cdot (1-(1-t_i)^n)\cdot t_{i+1}=p_{i+1}\cdot (1-(1-t_{i+1})^n)\cdot t_{i} & & & &\forall i\in [r-1] \\ 
\sum_{i\in [r]}t_i=1\end{cases}$$

Or equivalently:

$$\begin{cases} p_i\cdot \sum_{k=1}^{n-1}(1-t_i)^k\ \ \  =\ \ \  p_{i+1}\cdot \sum_{k=1}^{n-1}(1-t_{i+1})^k  & & & &\forall i\in [r-1] \\ 
\sum_{i\in [r]}t_i=1\end{cases}$$


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)?

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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][1] (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. 

  [1]: http://mathoverflow.net/questions/187396/does-this-equation-has-a-closed-form-solution-for-t-1-p-sum-i-0nti