Inequalities

problem
Near 10

Age

14 to 16

Challenge level

10 must remain within easy reach...
problem
Powerful order

Age

14 to 16

Challenge level

Powers of numbers might look large, but which of these is the largest...
problem
Square mean

Age

14 to 16

Challenge level

Is the mean of the squares of two numbers greater than, or less than, the square of their means?
problem
Approximating pi

Age

14 to 18

Challenge level

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?
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.
problem
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?
problem
Integral inequality

Age

16 to 18

Challenge level

An inequality involving integrals of squares of functions.
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?
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?
problem
Which is cheaper?

Age

14 to 16

Challenge level

When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?