TUYMAADA 2007 Problem 4

Determine maximum real $k$ such that there exist a set $X$ and its subsets $Y_{1}$ , $Y_{2}$ , $...$ , $Y_{31}$ satisfying the following conditions: (1) for every two elements of $X$ there is an index $i$ such that $Y_{i}$ contains neither of these elements;

(2) if any non-negative numbers $\alpha_{i}$ are assigned to the subsets $Y_{i}$ and $\alpha_{1}+\dots+\alpha_{31}=1$ then there is an element $x\in X$ such that the sum of $\alpha_{i}$ corresponding to all the subsets $Y_{i}$ that contain $x$ is at least $k$ .