Simultaneous equations

  • Not a Polite Question
    problem

    Not a polite question

    Age
    11 to 14
    Challenge level
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    When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...
  • Coffee
    problem

    Coffee

    Age
    14 to 16
    Challenge level
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    To make 11 kilograms of this blend of coffee costs £15 per kilogram. The blend uses more Brazilian, Kenyan and Mocha coffee... How many kilograms of each type of coffee are used?
  • System Speak
    problem

    System speak

    Age
    16 to 18
    Challenge level
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    Five equations... five unknowns... can you solve the system?
  • Always Two
    problem

    Always two

    Age
    14 to 18
    Challenge level
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    Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

  • Quadratic Harmony
    problem

    Quadratic harmony

    Age
    16 to 18
    Challenge level
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    Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
  • Escriptions
    problem

    Escriptions

    Age
    16 to 18
    Challenge level
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    For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
  • Pair Squares
    problem

    Pair squares

    Age
    16 to 18
    Challenge level
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    The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.
  • Gift of Gems
    problem

    Gift of gems

    Age
    14 to 16
    Challenge level
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    Four jewellers share their stock. Can you work out the relative values of their gems?
  • Polycircles
    problem

    Polycircles

    Age
    14 to 16
    Challenge level
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    Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

  • Rudolff's Problem
    problem

    Rudolff's problem

    Age
    14 to 16
    Challenge level
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    A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?