Solution
= Solution
Apply the <Szemerédi theorem> with density $\varepsilon$ and progression length five. For all sufficiently large $n$, the set $A$ contains a nonconstant five-term <arithmetic progression>
$$
e-2r,\quad e-r,\quad e,\quad e+r,\quad e+2r,
$$
where $r\ne0$. Set
$$
a=e-2r,
\quad b=e-r,
\quad c=e+r,
\quad d=e+2r.
$$
These four elements and $e$ are distinct, and direct addition gives
$$
a+b+c+d
=(e-2r)+(e-r)+(e+r)+(e+2r)
=4e.
$$
Solved by gpt-5.6-sol high.