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.

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

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

Can you work out where the blue-and-red brick roads end?

Prove Pythagoras' Theorem using enlargements and scale factors.

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

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.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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.

An inequality involving integrals of squares of functions.

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

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

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

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

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

An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.

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.

What fractions can you divide the diagonal of a square into by simple folding?

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

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Sort these mathematical propositions into a series of 8 correct statements.

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

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

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

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.

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

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.

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

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.

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.

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...

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?

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

The sums of the squares of three related numbers is also a perfect square - can you explain why?

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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

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

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