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Broad Topics > sfh10 > Stage 5 needs dealing with

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So Big

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

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Orthogonal Circle

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

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30-60-90 Polypuzzle

Stage: 5 Challenge Level: Challenge Level:1

Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.

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Escriptions

Stage: 5 Challenge Level: Challenge Level:1

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

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Strange Rectangle 2

Stage: 5 Challenge Level: Challenge Level:1

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

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Square Pair Circles

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Incircles

Stage: 5 Challenge Level: Challenge Level:1

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

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Telescoping Functions

Stage: 5

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.

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Gosh Cosh

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.

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One Basket or Group Photo

Stage: 2, 3, 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Libby Jared helped to set up NRICH and this is one of her favourite problems. It's a problem suitable for a wide age range and best tackled practically.

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W Mates

Stage: 5 Challenge Level: Challenge Level:1

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

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Flexi Quad Tan

Stage: 5 Challenge Level: Challenge Level:1

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

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Napoleon's Hat

Stage: 5 Challenge Level: Challenge Level:1

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?

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Mechanical Integration

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Polynomial Relations

Stage: 5 Challenge Level: Challenge Level:1

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

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Power Up

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

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Balances and Springs

Stage: 4 Challenge Level: Challenge Level:1

Balancing interactivity with springs and weights.

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Telescoping Series

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

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Weekly Challenge 49: Get in Line

Stage: 4 and 5 Challenge Level: Challenge Level:1

Match the graphs, the processes and the equations.

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Magical Maze - 35 Activities

Stage: 4 and 5

Investigations and activities for you to enjoy on pattern in nature.

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Flexi Quads

Stage: 5 Challenge Level: Challenge Level:1

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

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Cushion Ball

Stage: 5 Challenge Level: Challenge Level:1

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

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Stringing it Out

Stage: 4 Challenge Level: Challenge Level:1

Explore the transformations and comment on what you find.

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Scientific Curves

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you sketch these difficult curves, which have uses in mathematical modelling?

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Amida

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

To draw lots each player chooses a different upright, the paper is then unrolled, the paths charted and the results declared. Prove that no two paths ever end up at the foot of the same upright?

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Spring Frames

Stage: 5 Challenge Level: Challenge Level:1

Find the equilibrium points for these configurations of springs.

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Maltese Cross

Stage: 5 Challenge Level: Challenge Level:1

Sketch the graph of $xy(x^2 - y^2) = x^2 + y^2$ consisting of four curves and a single point at the origin. Convert to polar form. Describe the symmetries of the graph.

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Rain or Shine

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

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Witch of Agnesi

Stage: 5 Challenge Level: Challenge Level:1

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.