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Numbers from 1 to 15 with rare properties

Source: 2015 Peru Cono Sur TST P2

July 10, 2023

number theory

Problem Statement

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.

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