Given quadratic forms $f_k(x) = \sum_{i,j=1}^n a_{ij,k}x_ix_j$, the find necessary and sufficient conditions on $a_{ij,k}$ so that $f_k(x) \in [0,1]$ and $\sum_{k=1}^n f_k(x) = 1$ for all $x = (x_1,\dots, x_n)$ with $x_i\ge0$ and $\sum_{i = 1}^n x_i = 1$.
For $n=2$, after some calculations, I have found the condition that $a_{ii,k} \in [0,1]$, $\sum_{k=1}^n a_{ij,k} = 1$ and $a_{ij,k} \in \left[-\sqrt{a_{ii,k}a_{jj,k}}, 1+\sqrt{(1-a_{ii,k})(1-a_{jj,k})}\right]$ but having difficulties for the case $n \ge 3$.