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.

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?

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. . . .

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

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 ?

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. . . .

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.

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

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?

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

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

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

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

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

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.

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

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.

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

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.

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

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. . . .

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

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

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

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

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?

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

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?

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

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?

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?

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.

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.

Find a quadratic formula which generalises Pick's Theorem.

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.