Challenge Level

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.

Challenge Level

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.

Challenge Level

Do you have enough information to work out the area of the shaded quadrilateral?

Challenge Level

Can you find the areas of the trapezia in this sequence?

Challenge Level

Can you make sense of the three methods to work out the area of the kite in the square?

Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors.

Challenge Level

Can you make sense of these three proofs of Pythagoras' Theorem?

Challenge Level

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

Challenge Level

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

Challenge Level

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Challenge Level

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

Challenge Level

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

Challenge Level

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

Challenge Level

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?

Challenge Level

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

Challenge Level

Four jewellers share their stock. Can you work out the relative values of their gems?

Challenge Level

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. . . .

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

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!

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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.

Challenge Level

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.

Challenge Level

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Challenge Level

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?

Challenge Level

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?

Challenge Level

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

Challenge Level

Can you rearrange the cards to make a series of correct mathematical statements?

Challenge Level

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

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.

Challenge Level

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

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

Challenge Level

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

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

Challenge Level

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

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.

Challenge Level

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.

Challenge Level

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

Challenge Level

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.

Challenge Level

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.

Challenge Level

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?

Challenge Level

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.

Challenge Level

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.

Challenge Level

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

Challenge Level

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

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

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.