Solve the system of equations: ab = 1 bc = 2 cd = 3 de = 4 ea = 6
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.
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.
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.
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. . . .
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 ?
Can you find the values at the vertices when you know the values on
the edges of these multiplication arithmagons?
Change one equation in this pair of simultaneous equations very
slightly and there is a big change in the solution. Why?
The challenge is to find the values of the variables if you are to
solve this Sudoku.
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. . . .
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. . . .
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
Find the vertices of a pentagon given the midpoints of its sides.
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?
Which is bigger, n+10 or 2n+3? Can you find a good method of
answering similar questions?
Find a quadratic formula which generalises Pick's Theorem.
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.
You need to find the values of the stars before you can apply normal Sudoku rules.
Four numbers on an intersection that need to be placed in the
surrounding cells. That is all you need to know to solve this
A Sudoku with a twist.
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.
How many intersections do you expect from four straight lines ?
Which three lines enclose a triangle with negative co-ordinates for
every point ?
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
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the 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?
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
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?
When I park my car in Mathstown, there are two car parks to choose
from. Which car park should I use?
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
Crack this code which depends on taking pairs of letters and using
two simultaneous relations and modulus arithmetic to encode the
A simple method of defining the coefficients in the equations of chemical reactions with the help of a system of linear algebraic equations.
Find out how to model a battery mathematically
Investigate the effects of the half-lifes of the isotopes of cobalt
on the mass of a mystery lump of the element.