If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

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 bag contains red and blue balls. You are told the probabilities of drawing certain combinations of balls. Find how many red and how many blue balls there are in the bag.

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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. . . .

Change one equation in this pair of simultaneous equations very slightly and there is a big change in the solution. Why?

Use graphs to gain insights into an area and perimeter problem, or use your knowledge of area and perimeter of rectangles to gain insight into the graphs.

Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7

Solve the equations to identify the clue numbers in this Sudoku problem.

How many intersections do you expect from four straight lines ? Which three lines enclose a triangle with negative co-ordinates for every point ?

There are lots of different methods to find out what the shapes are worth - how many can you find?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

You need to find the values of the stars before you can apply normal Sudoku rules.

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.

When I park my car in Mathstown, there are two car parks to choose from. Which car park should I use?

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 ?

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

The challenge is to find the values of the variables if you are to solve this Sudoku.

To make 11 kilograms of this blend of coffee costs £15 per kilogram. The blend uses more Brazilian, Kenyan and Mocha coffee... How many kilograms of each type of coffee are used?

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.

Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one apiece to the rest in token of regard; and they. . . .

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

Solve the system of equations: ab = 1 bc = 2 cd = 3 de = 4 ea = 6

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

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.

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

Find a quadratic formula which generalises Pick's Theorem.

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

Plane 1 contains points A, B and C and plane 2 contains points A and B. Find all the points on plane 2 such that the two planes are perpendicular.

A simple method of defining the coefficients in the equations of chemical reactions with the help of a system of linear algebraic equations.

Investigate the effects of the half-lifes of the isotopes of cobalt on the mass of a mystery lump of the element.