Reasoning, convincing and proving

  • Rational Round
    problem

    Rational round

    Age
    16 to 18
    Challenge level
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    Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.
  • Leonardo's Problem
    problem

    Leonardo's problem

    Age
    14 to 18
    Challenge level
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    A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?
  • Sixty-Seven Squared
    problem

    Sixty-seven squared

    Age
    16 to 18
    Challenge level
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    Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?
  • Eyes Down
    problem

    Eyes down

    Age
    16 to 18
    Challenge level
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    The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?
  • Dalmatians
    problem

    Dalmatians

    Age
    14 to 18
    Challenge level
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    Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.
  • Make 37 Poster
    problem

    Make 37

    Age
    5 to 11
    Challenge level
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    Four bags contain a large number of 1s, 3s, 5s and 7s. Can you pick any ten numbers from the bags so that their total is 37?

  • Königsberg
    problem

    Königsberg

    Age
    11 to 14
    Challenge level
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    Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

  • Take Three From Five
    problem

    Take three from five

    Age
    11 to 16
    Challenge level
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    Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

  • Top-Heavy Pyramids
    problem

    Top-heavy pyramids

    Age
    11 to 14
    Challenge level
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    Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
  • Terminology
    problem

    Terminology

    Age
    14 to 16
    Challenge level
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    Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?