Reasoning, convincing and proving

  • Converse
    problem

    Converse

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
  • Three Ways
    problem

    Three ways

    Age
    16 to 18
    Challenge level
    filled star empty star empty star
    If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
  • Common Divisor
    problem

    Common divisor

    Age
    14 to 16
    Challenge level
    filled star empty star empty star
    Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
  • Long Short
    problem

    Long short

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
  • Mod 3
    problem

    Mod 3

    Age
    14 to 16
    Challenge level
    filled star empty star empty star
    Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
  • Our Ages
    problem

    Our ages

    Age
    14 to 16
    Challenge level
    filled star empty star empty star
    I am exactly n times my daughter's age. In m years I shall be ... How old am I?
  • DOTS Division
    problem

    DOTS division

    Age
    14 to 16
    Challenge level
    filled star filled star filled star

    Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

  • There's a limit
    problem

    There's a limit

    Age
    14 to 18
    Challenge level
    filled star empty star empty star
    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
    filled star filled star filled star
    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
    filled star empty star empty star
    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.