Problem 1

Prove that the set

\{1, 2, \ldots, 1989\}

can be expressed as the union of

117

pairwise disjoint subsets

A_1, A_2, \ldots, A_{117}

, each containing

17

elements, such that the sum of the elements is the same in every

A_i

Step 4 of 5: Give the leftover number 995 a home

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In plain words

The odd one out, $995$ , simply borrows a ready-made balanced pair to sit with, so it doesn't spoil the pattern.

\{995\} \cup \{r_0, 1990-r_0\} \quad\text{sums to}\quad 995 + 1990 = 2985

Detailed analysis

Take one more untouched pair $\{r_0,1990-r_0\}$ from Step 2 and combine it with the leftover $995$ to form a $117$ th block $\{995,r_0,1990-r_0\}$ ; its sum is $995+1990=2985$ , matching every triple from Step 3. Now all $117$ blocks are $3$ -element sets with the same sum $2985$ , built from $175$ of the $994$ pairs plus the number $995$ .