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

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

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.

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.

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.

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

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

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.

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

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

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

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.

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

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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?

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

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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

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

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

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.

What is the largest number of intersection points that a triangle and a quadrilateral can have?

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?

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

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

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

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?

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.

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. . . .

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

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.

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

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.

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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

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

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.

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.

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.

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.

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

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