Question 87
Call the set $\{a_1, a_2, \dots, a_n\}$ of positive integers good if
$$
\gcd(a_k, \sum_{i \neq k} a_i) \gt 1
$$
for each $k = \overline{1, n}$.
Prove that for each $c \in (0, 1)$ there exists a positive integer $M$ such that for any positive integer $K \ge M$ one can find a good set with at least $cK$ elements all elements of which doesn't exceed $K$.
(Yahor Dubovik)
$$
\gcd(a_k, \sum_{i \neq k} a_i) \gt 1
$$
for each $k = \overline{1, n}$.
Prove that for each $c \in (0, 1)$ there exists a positive integer $M$ such that for any positive integer $K \ge M$ one can find a good set with at least $cK$ elements all elements of which doesn't exceed $K$.
(Yahor Dubovik)