Zhendarov R.g.

A segment of unit length is divided into 11 segments, the length of each of which does not exceed $a$. For what values of $a$ can it be asserted that from any three of the resulting segments, a triangle can be formed?
It is impossible to divide into segments smaller than $\frac{1}{11}$, so $a \geq \frac{1}{11}$. If $a \geq 1 / 10$, a counterexample is a division into nine segments of length $1 / 10$ and two of length $1 / 20$.

## Answer

For $\frac{1}{11} \leq a<1 / 10$.