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?

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.

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

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

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?

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?

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?

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. . . .

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?

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

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. . . .

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.

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.

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?

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

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

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

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.

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

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

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

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

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?

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

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

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 make sense of these three proofs of Pythagoras' Theorem?

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

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

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

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.

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}.

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

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

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

Can you find a rule which relates triangular numbers to square numbers?

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.

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?

Can you find a rule which connects consecutive triangular numbers?

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.

Show that all pentagonal numbers are one third of a triangular number.

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

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.