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 1 of 5: Bound the two largest pair-sums

Previous step Next step

In plain words

The two sums built with the largest element are squeezed strictly between half of sA and sA, so neither can divide it.

\dfrac{s_A}{2} < a_2+a_4,\ a_3+a_4 < s_A

Detailed analysis

Order the entries so that $a_1<a_2<a_3<a_4$ . From $a_2>a_1$ and $a_4>a_3$ we get $2(a_2+a_4)>s_A$ , and from $a_3>a_1$ and $a_4>a_2$ we get $2(a_3+a_4)>s_A$ ; both $a_2+a_4$ and $a_3+a_4$ are also clearly less than $s_A$ . A number strictly between $s_A/2$ and $s_A$ can never divide $s_A$ , so neither of these two sums divides $s_A$ , leaving only the four sums $a_1+a_2,\ a_1+a_3,\ a_1+a_4,\ a_2+a_3$ as possible divisors of $s_A$ ; hence $n_A\le4$ .