Show that there exists $i\in \lbrace 1, 2, 3 \rbrace $ s.t. there exists $a, b\in A_i $ s.t. $a+b\in B $. Let $A=\lbrace 1, 2, 3,..., 2019\rbrace= A_1\cup A_2\cup A_3$, where $A_1\cap A_2=A_2\cap A_3= A_1\cap A_3=\emptyset $ and $B=\lbrace 672, 1008, 1344, 1680, 2016\rbrace $. 
Show that there exists $i\in \lbrace 1, 2, 3 \rbrace $ s.t. there exists $a, b\in A_i , a\neq b$ s.t. $a+b\in B $. 
I consider $A_1=\lbrace x_1, x_2, ..., x_k \rbrace$. Then $2018-x_1, 2018-x_2,..., 2018-x_k\notin A_1$.But is not enough.

Notice that any two of {840,504,168,1176} are not allowed to be in the same $A_i$, then there is a contradiction.
