Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

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.

Can you work out the fraction of the original triangle that is covered by the green triangle?

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

Can you work out the fraction of the original triangle that is covered by the inner triangle?

Move the point P to see how P' moves. Then use your insights to calculate a missing length.

A trapezium is divided into four triangles by its diagonals. Can you work out the area of the trapezium?

Can you find an efficent way to mix paints in any ratio?

A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.

Using an understanding that 1:2 and 2:3 were good ratios, start with a length and keep reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range.

The Pythagoreans noticed that nice simple ratios of string length made nice sounds together.

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.

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

What's the most efficient proportion for a 1 litre tin of paint?

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

Why is the modern piano tuned using an equal tempered scale and what has this got to do with logarithms?

Match pairs of cards so that they have equivalent ratios.

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

Which exact dilution ratios can you make using only 2 dilutions?

What does the empirical formula of this mixture of iron oxides tell you about its consituents?

Scientists often require solutions which are diluted to a particular concentration. In this problem, you can explore the mathematics of simple dilutions