Rational Numbers | Singapore Mathematics Olympiad, 2013

Find the number of positive integer pairs (a,b) satisfying \(a^2 + b^2<2013\) and \(a^{2} b |(b^3 - a^3\)

We can start this sum by rearranging the given values :

Let \( k = \frac { b^3 - a^3 }{a^{2}b} \)

Again we can write it like : \( k = (\frac {b}{a})^2 - \frac {a}{b} \)

Try to use this value and then try to do the rest of the sum.......

From the first hint we can say :

\((\frac {a}{b})^{3} + k (\frac {a}{b})^2 - 1 = 0\)

The only possible positive rational number solution of \(x^3 +kx^2 -1 = 0\) is x = 1 namely a = b . Conversely , if a = b then it is obvious that \(a^2b |(b^3 - a^3\)

Then 2013 > \(a^2 +b^2 = 2a^2 \) implies \(a\leq 31 \). (Answer )

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