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Dividing into Two pair with Sum n

Source: 1992 Japan Mathematical Olympiad Finals, Problem 5

November 2, 2009
number theory proposednumber theory

Problem Statement

Suppose that n≥2 n\geq 2 be integer and a1, a2, a3, a4 a_1,\ a_2,\ a_3,\ a_4 satisfy the following condition: i)  n \ n and a_i\ (i \equal{} 1,\ 2,\ 3,\ 4) are relatively prime. ii) \ (ka_1)_n \plus{} (ka_2)_n \plus{} (ka_3)_n \plus{} (ka_4)_n \equal{} 2n for k \equal{} 1,\ 2,\ \cdots ,\ n \minus{} 1. Note that (a)n (a)_n expresses the divisor when a a is divided by n n. Prove that (a1)n, (a2)n, (a3)n, (a4)n (a_1)_n,\ (a_2)_n,\ (a_3)_n,\ (a_4)_n can be divided into two pair with sum n n.
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