Creating and manipulating expressions and formulae

  • AMGM
    problem

    AMGM

    Age
    14 to 16
    Challenge level
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    Can you use the diagram to prove the AM-GM inequality?

  • Hot Pursuit
    problem

    Hot pursuit

    Age
    11 to 14
    Challenge level
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    I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...
  • Look before you leap
    problem

    Look before you leap

    Age
    16 to 18
    Challenge level
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    Relate these algebraic expressions to geometrical diagrams.
  • Around and Back
    problem

    Around and back

    Age
    14 to 16
    Challenge level
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    A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns around and heads back to the starting point where he meets the runner who is just finishing his first circuit. Find the ratio of their speeds.
  • Adding in Rows
    problem

    Adding in rows

    Age
    11 to 14
    Challenge level
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    List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
  • Little and Large
    problem

    Little and large

    Age
    16 to 18
    Challenge level
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    A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?
  • Why 24?
    problem

    Why 24?

    Age
    14 to 16
    Challenge level
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    Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
  • What's Possible?
    problem

    What's possible?

    Age
    14 to 16
    Challenge level
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    Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

  • Quick Times
    problem

    Quick times

    Age
    11 to 14
    Challenge level
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    32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.
  • Fair Shares?
    problem

    Fair shares?

    Age
    14 to 16
    Challenge level
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    A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?