Inequalities

problem
Favourite

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?
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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?
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Favourite

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?
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Favourite

Which Is Bigger?

Age

14 to 16

Challenge level

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?
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Favourite

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?
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Favourite

Tet-Trouble

Age

14 to 16

Challenge level

Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units?
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Favourite

Unit Interval

Age

14 to 18

Challenge level

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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Favourite

Quadratic Harmony

Age

16 to 18

Challenge level

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
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Favourite

Giants

Age

16 to 18

Challenge level

Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?
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Favourite

Eyes Down

Age

16 to 18

Challenge level

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?