MathLabs

Problem 1

Prove that the set {1,2,…,1989}\{1, 2, \ldots, 1989\} can be expressed as the union of 117117 pairwise disjoint subsets A1,A2,…,A117A_1, A_2, \ldots, A_{117}, each containing 1717 elements, such that the sum of the elements is the same in every AiA_i.
Step 2 of 5: Pair every number with its mirror image around 995
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In plain words

Picture the numbers arranged symmetrically around 995995 on a number line: folding the line at 995995 matches up rr with 1990−r1990-r perfectly, leaving the fold point itself unmatched.

{1,…,1989}={995}∪⋃r=1994{r, 1990−r}\{1,\ldots,1989\} = \{995\} \cup \bigcup_{r=1}^{994} \{r,\ 1990-r\}
Detailed analysis

Besides the middle number 995995, every other number r∈{1,…,1989}r\in\{1,\ldots,1989\} has a partner 1990−r1990-r also in the range, and r+(1990−r)=1990r+(1990-r)=1990. This splits the remaining 19881988 numbers into 994994 pairs, each of sum 19901990, i.e. exactly twice the average 995995. Adding a whole pair to any collection changes its sum by the fixed amount 19901990, which is the key tool for balancing the 117117 blocks later.