Creating and manipulating expressions and formulae

problem
Favourite

Quadratic Harmony

Age

16 to 18

Challenge level

1 out of 3

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.
problem
Favourite

System Speak

Age

16 to 18

Challenge level

1 out of 3

Five equations... five unknowns... can you solve the system?
problem
Favourite

Sums of Squares

Age

16 to 18

Challenge level

1 out of 3

Can you prove that twice the sum of two squares always gives the sum of two squares?
problem
Favourite

Binomial

Age

16 to 18

Challenge level

2 out of 3

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
problem
Favourite

Mechanical Integration

Age

16 to 18

Challenge level

2 out of 3

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
problem

Absurdity Again

Age

16 to 18

Challenge level

1 out of 3

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
problem

Old Nuts

Age

16 to 18

Challenge level

1 out of 3

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?
problem

Incircles

Age

16 to 18

Challenge level

1 out of 3

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?
problem

Reciprocals

Age

16 to 18

Challenge level

1 out of 3

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.
problem

Plum Tree

Age

14 to 18

Challenge level

1 out of 3

Label this plum tree graph to make it totally magic!