Problem 1.
Determine all composite integers $n>1$ that satisfy the following property: if $d_1, d_2, \ldots, d_k$ are all the positive divisors of $n$ with $1=d_1<d_2<\cdots<d_k=n$, then $d_i$ divides $d_{i+1}+d_{i+2}$ for every $1 \leqslant i \leqslant k-2$.

proposed by Santiago Rodriguez, Colombia