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Disjoint Pairs

Source: USAMO 1998

October 9, 2005
modular arithmeticnumber theory proposednumber theory

Problem Statement

Suppose that the set {1,2,⋯ ,1998}\{1,2,\cdots, 1998\} has been partitioned into disjoint pairs {ai,bi}\{a_i,b_i\} (1≤i≤9991\leq i\leq 999) so that for all ii, ∣ai−bi∣|a_i-b_i| equals 11 or 66. Prove that the sum ∣a1−b1∣+∣a2−b2∣+⋯+∣a999−b999∣ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| ends in the digit 99.
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