Inequalities

problem
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

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

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.
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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

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
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?
problem
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?
problem
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?
problem
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?
problem
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 ?