Creating and manipulating expressions and formulae

  • Binomial
    problem
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    Binomial

    Age
    16 to 18
    Challenge level

    By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

  • Mechanical Integration
    problem
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    Mechanical Integration

    Age
    16 to 18
    Challenge level

    To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

  • Complex partial fractions
    problem
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    Complex Partial Fractions

    Age
    16 to 18
    Challenge level

    To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.

  • Absurdity Again
    problem

    Absurdity Again

    Age
    16 to 18
    Challenge level
    What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
  • Old Nuts
    problem

    Old Nuts

    Age
    16 to 18
    Challenge level
    In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?
  • Incircles
    problem

    Incircles

    Age
    16 to 18
    Challenge level
    The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?
  • Reciprocals
    problem

    Reciprocals

    Age
    16 to 18
    Challenge level
    Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.
  • Plum Tree
    problem

    Plum Tree

    Age
    14 to 18
    Challenge level
    Label this plum tree graph to make it totally magic!
  • W Mates
    problem

    W Mates

    Age
    16 to 18
    Challenge level
    Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.
  • Algebra from Geometry
    problem

    Algebra From Geometry

    Age
    11 to 16
    Challenge level
    Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
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