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 4 of 5: Pin down n and m

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In plain words

Positivity of a1 turns into a size restriction on n and m, which together with n<m leaves only two possibilities.

2a_1=(a_1+a_3)+(a_1+a_2)-(a_2+a_3)=s_A\left(\dfrac1n+\dfrac1m-\dfrac12\right)

Detailed analysis

Combining the three equations gives $2a_1=(a_1+a_3)+(a_1+a_2)-(a_2+a_3)=s_A\left(\dfrac1n+\dfrac1m-\dfrac12\right)$ . Since $a_1>0$ , this forces $\dfrac1n+\dfrac1m>\dfrac12$ . If $n\ge4$ then $\dfrac1n\le\dfrac14$ , which would force $1/m>1/4$ , i.e. $m<4\le n$ , contradicting $m>n$ ; hence $n=3$ . Then $\dfrac1m>\dfrac12-\dfrac13=\dfrac16$ gives $m<6$ , and together with $m>n=3$ this leaves $m\in\{4,5\}$ .