IMO 1977 LL ROM37

Let A1, A2, . . . , An+1 be positive integers such that (Ai, An+1)

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IMO 1977 LL ROM37

Origin: ROM

Problem

Let A1, A2, . . . , An+1 be positive integers such that (Ai, An+1) = 1 for every i = 1, 2, . . ., n. Show that the equation xA1

  • xA2
  • \cdot \cdot \cdot + xAn n = xAn+1 n+1 has an infinite set of solutions (x1, x2, . . . , xn+1) in positive integers.

Solution

We look for a solution with xA1 = \cdot \cdot \cdot = xAn n = nA1A2\cdot\cdot\cdotAnx and xn+1 = ny. In order for this to be a solution we must have A1A2 \cdot \cdot \cdot Anx + 1 =

An+1y. This equation has infinitely many solutions (x, y) in N, since A1A2 \cdot \cdot \cdot An and An+1 are coprime.