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

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 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.

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

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 all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

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

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

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

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

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?

An algebra 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.

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

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.

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.

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. . . .

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?

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

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

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

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.

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

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

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

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 ?

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

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

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

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

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.

Relate these algebraic expressions to geometrical diagrams.

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...

Can you find the lap times of the two cyclists travelling at constant speeds?

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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?

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

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

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

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

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.