Problem 1

Given any set

A=\{a_1,a_2,a_3,a_4\}

of four distinct positive integers, we denote the sum

a_1+a_2+a_3+a_4

s_A

. Let

n_A

denote the number of pairs

(i,j)

with

1\le i<j\le4

for which

a_i+a_j

divides

s_A

. Find all sets

A

of four distinct positive integers which achieve the largest possible value of

n_A

Step 3 of 5: Parametrize the remaining two sums

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In plain words

The two smaller divisors of sA must be strictly smaller than sA/2, and they are ordered the same way as the numbers they come from.

a_1+a_3=\dfrac{s_A}{n},\quad a_1+a_2=\dfrac{s_A}{m}

Detailed analysis

Since $a_3<a_4$ , $a_1+a_3 < a_1+a_4 = s_A/2$ ; writing $a_1+a_3=\dfrac{s_A}{n}$ therefore forces $n>2$ , i.e. $n\ge3$ . Likewise $a_2<a_3$ gives $a_1+a_2<a_1+a_3$ , so writing $a_1+a_2=\dfrac{s_A}{m}$ forces $m>n$ .