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

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.

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

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

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?

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

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

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 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?

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

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. . . .

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.

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?

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

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?

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?

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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.

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

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

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

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.

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

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.

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

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

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.

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.

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.

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

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

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?

Relate these algebraic expressions to geometrical diagrams.

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

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

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

Can you find a rule which connects consecutive triangular numbers?

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?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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}.

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

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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