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Singapore Math Olympiad

Rational Numbers | Singapore Mathematics Olympiad, 2013

Try this beautiful problem from Singapore Mathematics Olympiad based on rational numbers. You may use sequential hints to solve the problem.

Try this beautiful problem from Singapore Mathematics Olympiad based on rational numbers.

Problem – Rational Numbers (SMO )


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

  • 30
  • 32
  • 31
  • 29

Key Concepts


Number Theory

Rational Number

Analysis of Numbers

Check the Answer


But try the problem first…

Answer: 31

Source
Suggested Reading

Singapore Mathematics Olympiad – 2013 – Senior Section – Problem No. 18

Challenges and Thrills – Pre – College Mathematics

Try with Hints


First hint

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…….

Second Hint

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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