2011 IMO Shortlist C6

Let $n$ be a positive integer and let $W=\ldots x_{-1} x_{0} x_{1} x_{2} \ldots$ be an infinite periodic word consisting of the letters $a$ and $b$. Suppose that the minimal period $N$ of $W$ is greater than $2^{n}$. A finite nonempty word $U$ is said to appear in $W$ if there exist indices $k \leq \ell$ such that $U=x_{k} x_{k+1} \ldots x_{\ell}$. A finite word $U$ is called ubiquitous if the four words $U a, U b, a U$, and $b U$ all appear in $W$. Prove that there are at least $n$ ubiquitous finite nonempty words.

设$n$为正整数，设$W=\ldots x_{-1} x_{0} x_{1} x_{2} \ldots$为由字母$a$和$b$组成的无限周期词。假设$W$的最小周期$N$大于$2^{n}$。如果存在索引 $k \leq \ell$ 使得 $U=x_{k} x_{k+1} \ldots x_{\ell}$，则称有限非空词 $U$ 出现在 $W$ 中。如果四个词 $U a、U b、a U$ 和 $b U$ 都出现在 $W$ 中，则有限词 $U$ 被称为无处不在。证明至少有 $n$ 个普遍存在的有限非空词。