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.

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.

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.

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

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.

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

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?

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

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

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

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}.

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

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.

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

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

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.

If you know the perimeter of a right angled triangle, what can you say about the area?

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

Can you find a rule which connects consecutive triangular numbers?

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

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

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.

There are unexpected discoveries to be made about square numbers...

What is special about the difference between squares of numbers adjacent to multiples of three?

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 explain what is going on in these puzzling number tricks?

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

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

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 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.

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

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

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

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. . . .

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

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

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?

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?

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .