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.

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.

Relate these algebraic expressions to geometrical diagrams.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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

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.

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

Can you work through these direct proofs, using our interactive proof sorters?

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

If you think that mathematical proof is really clearcut and universal then you should read this article.

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

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

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?

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

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. . . .

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

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

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

Have a go at being mathematically negative, by negating these statements.

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

Explore a number pattern which has the same symmetries in different bases.

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.