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Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

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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?

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The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. . . .

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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?

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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.

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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.

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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. . . .

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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.

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Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

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Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

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

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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.

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Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

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Find the area of the shaded region created by the two overlapping triangles in terms of a and b?

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

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

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

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P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

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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.

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

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Construct a line parallel to one side of a triangle so that the triangle is divided into two equal areas.

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

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Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

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

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

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

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

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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. . . .

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One night two candles were lit. Can you work out how long each candle was originally?

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Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.

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

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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. . . .

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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. . . .

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A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.

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

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

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Andy is desperate to reach John o'Groats first. Can you devise a winning race plan?

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

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

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

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Which dilutions can you make using 10ml pipettes and 100ml measuring cylinders?

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

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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.

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A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

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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.

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

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A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?