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 2 of 5: Force the two middle sums to split sA evenly

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In plain words

If a1+a4 divides sA at all, the only way the complementary sum a2+a3 can also divide sA is if the two split sA exactly in half.

a_1+a_4=a_2+a_3=\dfrac{s_A}{2}

Detailed analysis

Achieving $n_A=4$ requires $a_1+a_2,a_1+a_3,a_1+a_4,a_2+a_3$ to all divide $s_A$ . Put $d=a_1+a_4$ ; since $d\mid s_A$ , write $s_A=dt$ with $t\ge2$ . Then $a_2+a_3=s_A-d=d(t-1)$ , and for this to also divide $s_A$ we need $d(t-1)\mid dt$ , i.e. $(t-1)\mid t$ , i.e. $(t-1)\mid1$ , which forces $t=2$ . Hence $a_1+a_4=a_2+a_3=\dfrac{s_A}{2}$ .