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

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 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 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.

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

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 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. . . .

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

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.

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

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.

Relate these algebraic expressions to geometrical diagrams.

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.

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

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

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

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

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

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}.

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 ...?

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

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.

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.

Show that all pentagonal numbers are one third of a triangular 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?

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 a rule which connects consecutive triangular numbers?

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

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

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

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

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

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

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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. . . .

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

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

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. . . .

Can you explain what is going on in these puzzling number tricks?

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

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?

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.