A nice inequality with positive real tuples I have found out this very nice inequality, which I have not been able to solve and I think it is an interesting challenge. Simulation results seem to verify it.  

Consider $n$ positive (strictly larger than zero) real numbers $x_1,x_2,...,x_n$ such that 
  $x_1+x_2+...+x_n=1$ and $n$ other arbitrary positive (strictly larger than zero) real numbers, denoted as $y_1,y_2,...y_n$.
  Show that:
   $$\left(\frac{1}{\sum_{i=1}^n{\frac{x_i}{y_i}}}+1\right) \cdot \sum_{i=1}^n{\frac{x_i}{y_i+1}} \leq 1.$$

Let me know if you can help me. 
 A: We need to prove that
$$\left(\frac{1}{\sum\limits_{i=1}^n{\frac{x_i}{y_i}}}+1\right) \sum_{i=1}^n{\frac{x_i}{y_i+1}} \leq 1$$ or
$$\sum_{i=1}^n{\frac{x_i}{y_i+1}}\leq\frac{\sum\limits_{i=1}^n\frac{x_i}{y_1}}{1+\sum\limits_{i=1}^n\frac{x_i}{y_1}}$$ or
$$\sum_{i=1}^n\left(\frac{x_i}{y_i+1}-x_i\right)\leq\frac{\sum\limits_{i=1}^n\frac{x_i}{y_1}}{1+\sum\limits_{i=1}^n\frac{x_i}{y_1}}-1$$ or
$$\sum\limits_{i=1}^n\frac{x_iy_i}{y_i+1}\geq\frac{1}{1+\sum\limits_{i=1}^n\frac{x_i}{y_1}}$$ or
$$\left(1+\sum\limits_{i=1}^n\frac{x_i}{y_1}\right)\sum\limits_{i=1}^n\frac{x_iy_i}{y_i+1}\geq1$$ or
$$\sum\limits_{i=1}^n\left(\frac{x_i}{y_1}+x_i\right)\sum\limits_{i=1}^n\frac{x_iy_i}{y_i+1}\geq1$$ or
$$\sum\limits_{i=1}^n\frac{x_i(y_i+1)}{y_1}\sum\limits_{i=1}^n\frac{x_iy_i}{y_i+1}\geq1,$$
which is C-S:
$$\sum\limits_{i=1}^n\frac{x_i(y_i+1)}{y_1}\sum\limits_{i=1}^n\frac{x_iy_i}{y_i+1}\geq\left(\sum\limits_{i=1}^nx_i\right)^2=1.$$
Done!
A: The result follows directly from Jensen's inequality applied to the  function $u\mapsto u/(1+u),$ which is concave on $\mathbb R_+$.  Define a random variable $U$ for which $U=1/y_i$ with probability $x_i$, so for instance the expectation of $U$ is $EU=\sum_{i=1}^n x_i/y_i$ and $E(U/(1+U))= \sum_{i=1}^n x_i /(1+y_i)$ and  so on.  Then according to Jensen, $$E\frac U {1+U}\le \frac{EU}{1+EU},$$ which is equivalent to $$\left( \frac 1{EU}+1\right ) \frac{EU}{1+EU}\le 1,$$ which is the OP's original inequality.
