Mongolian National Mathematical Olympiad

First note that for any finite family of finite sets $\{X_i{\}}_{i\in I}$, we have $$|{\cup }_i{X}_i|\ge \sum _i|X_i|-\frac{1}{2}\sum _{i\ne j}|X_i\cap X_j|.$$ Indeed, if $x\in {\cup }_i{X}_i$ is contained in exactly $k$ of the sets $X_i$, then it is counted $k$ times in the first sum and $k(k-1)$ times in the second sum, and $1\ge k-\frac{k(k-1)}{2}$ for any integer $k\ge 1$.

Let $F=\{1,2,\dots ,2016\}$. For $a\in A$ and $n\in F$, let $$X_a^n=a+nB=\{a+nb(\mathrm{mod}2017)\mid b\in B\}.$$ Clearly, for $a\in A$ and $n\in F$, we have $|X_a^n|=|B|$. Moreover, for $a\ne a^{\prime }\in A$, we have $$|X_a^n\cap X_{a^{\prime }}^n|=\sum _{b,b^{\prime }\in B}{\delta }_{a+nb,a^{\prime }+n{b}^{\prime }}=\sum _{\begin{array}{c}b,b^{\prime }\in B\\ b\ne b^{\prime }\end{array}}{\delta }_{a+nb,a^{\prime }+n{b}^{\prime }}=\sum _{\begin{array}{c}b,b^{\prime }\in B\\ b\ne b^{\prime }\end{array}}{\delta }_{n,\frac{a-a^{\prime }}{b-b^{\prime }}}$$ using Kronecker's delta function notation. Summing over all $n\in F$, we get $$\sum _{n=1}^{2016}|X_a^n\cap X_{a^{\prime }}^n|=|B|(|B|-1).$$ Now let $X^n={\bigcup }_{a\in A}{X}_a^n=A+nB$. Then $$|X^n|\ge \sum _{a\in A}|X_a^n|-\frac{1}{2}\sum _{\begin{array}{c}a,a^{\prime }\in A\\ a\ne a^{\prime }\end{array}}|X_a^n\cap X_{a^{\prime }}^n|$$ and averaging over all $n\in F$, we get $$\begin{array}{rl}\frac{1}{|F|}\sum _{n\in F}|X^n|& \ge |A||B|-\frac{1}{2}\frac{|A|(|A|-1)|B|(|B|-1)}{|F|}\\ & =30\cdot 31-\frac{30\cdot 29\cdot 31\cdot 30}{2\cdot 2016}>729.\end{array}$$

It follows that there is $n\in F$ such that $|X^n|\ge 730$.