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.

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

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

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

Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?

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

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.

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?

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

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

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.

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.

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

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

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?

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?

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

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...

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.

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

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.

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 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.

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. . . .

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.

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

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

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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

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 ?

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.

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

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.

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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.

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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}.

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

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

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?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?