  We study a system of intervals $I_1,\ldots,I_k$ on the real line and a
continuous map $f$ with $f(I_1 \cup I_2 \cup \ldots \cup I_k)\supseteq I_1 \cup
I_2 \cup \ldots \cup I_k$. It's conjectured that there exists a periodic point
of period $\le k$ in $I_1\cup \ldots \cup I_k$. In this paper, we prove the
conjecture by a discretization method and reduce the initial problem to an
interesting combinatorial lemma concerning cyclic permutations. We also obtain
a non-concentration property of periodic points of small periods in intervals.
