Creating and manipulating expressions and formulae

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.
$Complex partial fractions$

problem
Favourite

Complex Partial Fractions

Age

16 to 18

Challenge level

2 out of 3

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.
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!
problem

W Mates

Age

16 to 18

Challenge level

1 out of 3

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.