Inequalities

  • Discrete Trends
    problem
    Favourite

    Discrete Trends

    Age
    16 to 18
    Challenge level

    Find the maximum value of n to the power 1/n and prove that it is a maximum.

  • Squareness
    problem
    Favourite

    Squareness

    Age
    16 to 18
    Challenge level

    The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

  • Random inequalities
    problem
    Favourite

    Random Inequalities

    Age
    16 to 18
    Challenge level

    Can you build a distribution with the maximum theoretical spread?

  • 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.
  • Classical Means
    problem

    Classical Means

    Age
    16 to 18
    Challenge level
    Use the diagram to investigate the classical Pythagorean means.
  • Fracmax
    problem

    Fracmax

    Age
    14 to 16
    Challenge level
    Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.
  • Jute bag with marbles of different colours spilling out.
    problem

    Inequalities

    Age
    11 to 14
    Challenge level

    A bag contains 12 marbles. There are more red than green but green and blue together exceed the reds. The total of yellow and green marbles is more than the total of red and blue. How many of each colour there are in the bag?

  • A gold gift box with a ribbon.
    problem

    Plutarch's Boxes

    Age
    11 to 14
    Challenge level

    According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

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