Let $a, b, c$ and $d$ be elements of the set $\{ 1, 2, 3,\ldots , 2014, 2015 \}$ such that $a < b < c < d$, $a + b$ is a divisor of $c + d$, and $a + c$ is a divisor of $b + d$. Determine the largest value that $a$ can take.
Source: 2015 Peru Cono Sur TST P2
Tags: number theory
Let $a, b, c$ and $d$ be elements of the set $\{ 1, 2, 3,\ldots , 2014, 2015 \}$ such that $a < b < c < d$, $a + b$ is a divisor of $c + d$, and $a + c$ is a divisor of $b + d$. Determine the largest value that $a$ can take.