Reasoning, convincing and proving

  • There's a limit
    problem

    There's a limit

    Age
    14 to 18
    Challenge level
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    Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
  • Diophantine n-tuples
    problem

    Diophantine n-tuples

    Age
    14 to 16
    Challenge level
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    Can you explain why a sequence of operations always gives you perfect squares?
  • Pythagorean Golden Means
    problem

    Pythagorean golden means

    Age
    16 to 18
    Challenge level
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    Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.
  • Euler's Squares
    problem

    Euler's squares

    Age
    14 to 16
    Challenge level
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    Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
  • Square Mean
    problem

    Square mean

    Age
    14 to 16
    Challenge level
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    Is the mean of the squares of two numbers greater than, or less than, the square of their means?
  • N000ughty thoughts
    problem

    N000ughty thoughts

    Age
    14 to 16
    Challenge level
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    How many noughts are at the end of these giant numbers?
  • Without Calculus
    problem

    Without calculus

    Age
    16 to 18
    Challenge level
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    Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.
  • Cosines Rule
    problem

    Cosines rule

    Age
    14 to 16
    Challenge level
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    Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.
  • Binomial
    problem

    Binomial

    Age
    16 to 18
    Challenge level
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    By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
  • Triangle Incircle Iteration
    problem

    Triangle incircle iteration

    Age
    14 to 16
    Challenge level
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    Keep constructing triangles in the incircle of the previous triangle. What happens?