Quadratic Equation | SMO, 2012 | Junior Section

Try this beautiful problem from Singapore Math Olympiad, 2012, Junior Senior based on Quadratic Equation.

Quadratic Equation - Singapore Mathematics Olympiad, 2012

Consider the equation

\(\sqrt {3x^2 - 8x + 1} + \sqrt {9x^2 - 24x - 8}\) = 3.

  • 9
  • 10
  • 11
  • 12

Key Concepts

Quadratic Function

Analysis of Number

Root of Equation

Check the Answer

Answer: 9

Singapore Mathematics Olympiad

Challenges and Thrills - Pre - College Mathematics

Try with Hints

As the first hint we can assume :

y =\(3x^2 - 8x + 1 \) then the equation becomes

y + \(\sqrt {3y^2 -11}\) = 3.

Lets try to do the rest of the sum ....................

If we are still stuck after the first hint we can say :

Then \( \sqrt {3y^2 - 11}\) = 3 - y.

Lets square the both sides , we have \( 3 y^2 - 11 = 9 -6y + y^2 \) ,

Then y = 2 or y = -5

Now solve \( 3x^2 - 8x +1 = 2^2 \)

Then x =3 and \(x = - 3^{-1}\)

Hence k = \(\frac {3}{3^{-1}}\) = 9 (Answer )

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Rational Numbers | Singapore Mathematics Olympiad, 2013

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

Answer: 31

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

Challenges and Thrills - Pre - College Mathematics

Try with Hints

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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