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

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

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

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

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.

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

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

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

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

An algebra task which depends on members of the group noticing the needs of others and responding.

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

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

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

By proving these particular identities, prove the existence of general cases.

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.

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

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?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Find the five distinct digits N, R, I, C and H in the following nomogram

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Can you find the value of this function involving algebraic fractions for x=2000?

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

Can you hit the target functions using a set of input functions and a little calculus and algebra?

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

Kyle and his teacher disagree about his test score - who is right?

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

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Relate these algebraic expressions to geometrical diagrams.

Can you find the lap times of the two cyclists travelling at constant speeds?

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.