MSF Challenge 2006
( A Joint Initiative with NIIT )
Mathematics with Computation
    
     
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Prestigious competition with attractive prizes.
Important Dates
This contest is open to schools in Delhi and NCR.. They may enter teams in two age groups
Group A (Junior Level): for students of classes IX and X

(Download Problems for Group A)
Submission of Form by School
July 5, 2006
Submission of Solution/Paper
July 5, 2006
Instructions
  • The problem you choose should be considered interesting to your team.
  • Mathematicians and Scientists often do the following activities to gain “intuition” about a problem situation. You may find them helpful too.
 
  • Try small cases or simpler versions of a problem first
  • Doing numerical computations
  • Drawing pictures
  • Writing programs or using computer software to list special cases or to test your hypothesis
  • Think a bit about whether your solution and method can generalize or can be used in modified situations. Mention these generalizations in your paper.
  • Even partial solutions may be acceptable. What is important is how you approach things, and whether your team got a few insights. Communicating those insights in the paper is important.
  • Think independently. Creative ideas and approaches will get more credit from our distinguished jury.

Problems
Group A (for Classes IX & X)
Problem A1:
Continued Fractions
Calculate the following sequence of fractions:

The above sequence of fractions can be continued (The three dots at the end mean “and so on”). There are a few things you can try to understand this sequence.

  • Find the next 5 terms.
  • Compute each fraction and represent as a rational number (as a ratio of integers).
  • Use a calculator or a computer program such as MS. Excel to calculate the first 8 digits of the decimal expansion of each fraction.

After experimenting with the solution, here are a few questions you can answer.

  • Can you predict the numerator and denominator of the fractions in this sequence? What is the numerator of the 100th fraction in the list? What is the denominator of the 99th fraction in the list?
  • What is the most efficient way of representing each fraction as a rational number? Can you use the rational expression of one fraction to find the next one efficiently? Can you find a rule (using symbols) to write a formula for computing the next fraction?
  • Is there a pattern in the decimal expansions of the fraction?
  • Can you write any fraction as a continued fraction? For example, the continued fraction of 5/3 is
  • What meaning can you ascribe to the symbol
where the three dots signify that the fraction continues forever.
 
Problem A2:
Let us save steel and money
You are a manufacturer of steel containers (for liquids) of two basic shapes – tins shaped as right circular cylinders and rectangular boxes. The containers’ capacity is 1 litre. Obviously you want to use the least amount of steel so as to maximize your profits. Can you work out the dimensions of the containers? How many different boxes of the same type will you have? Illustrate your response with graphs and tables, well documented.
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