IMO 1983 LL BUL13

Let p be a prime number and a1, a2, . . . , a(p+1)/2 different nat-

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IMO 1983 LL BUL13

Origin: BUL

Problem

Let p be a prime number and a1, a2, . . . , a(p+1)/2 different nat- ural numbers less than or equal to p. Prove that for each natural number r less than or equal to p, there exist two numbers (perhaps equal) ai and aj such that p \equivaiaj(mod r).