IMO 1970 LL SWE51

Let p be a prime number. A rational number x, with 0 < x < 1,

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IMO 1970 LL SWE51

Origin: SWE

Problem

Let p be a prime number. A rational number x, with 0 < x < 1, is written in lowest terms. The rational number obtained from x by adding p to both the numerator and the denominator differs from x by 1/p2. Determine all rational numbers x with this property.