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

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.

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

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.

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?

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. . . .

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.

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

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

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

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.

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

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?

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

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...

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

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?

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?

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

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.

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

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

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

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

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

Relate these algebraic expressions to geometrical diagrams.

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?

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

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.

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?

Can you explain why a sequence of operations always gives you perfect squares?

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

How good are you at finding the formula for a number pattern ?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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

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.

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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

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?

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.

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

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 ?

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...

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

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?

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