Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

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?

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

Keep constructing triangles in the incircle of the previous triangle. What happens?

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

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.

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Prove Pythagoras' Theorem using enlargements and scale factors.

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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.

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

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

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.

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. . . .

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

This is the second article on right-angled triangles whose edge lengths are whole numbers.

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

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

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.

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

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.

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.

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.

Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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

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?