MathDB

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

4