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### Orthogonal Circle

##### Stage: 5 Challenge Level:

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

### 30-60-90 Polypuzzle

##### Stage: 5 Challenge Level:

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

### So Big

##### Stage: 5 Challenge Level:

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

### Strange Rectangle 2

##### Stage: 5 Challenge Level:

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

### Escriptions

##### Stage: 5 Challenge Level:

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

### Square Pair Circles

##### Stage: 5 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.

### Incircles

##### Stage: 5 Challenge Level:

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 ...?

### Balances and Springs

##### Stage: 4 Challenge Level:

Balancing interactivity with springs and weights.

### W Mates

##### Stage: 5 Challenge Level:

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.

### Witch of Agnesi

##### Stage: 5 Challenge Level:

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

### Weekly Challenge 49: Get in Line

##### Stage: 4 and 5 Challenge Level:

Match the graphs, the processes and the equations.

### Mechanical Integration

##### Stage: 5 Challenge Level:

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

### Polynomial Relations

##### Stage: 5 Challenge Level:

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.

### Maltese Cross

##### Stage: 5 Challenge Level:

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.

### 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.

### One Basket or Group Photo

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

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.

##### Stage: 5 Challenge Level:

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.

### Cushion Ball

##### Stage: 5 Challenge Level:

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?

### Stringing it Out

##### Stage: 4 Challenge Level:

Explore the transformations and comment on what you find.

### Scientific Curves

##### Stage: 5 Challenge Level:

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

### Gosh Cosh

##### Stage: 5 Challenge Level:

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

### Power Up

##### Stage: 5 Challenge Level:

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

### Telescoping Series

##### Stage: 5 Challenge Level:

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

##### Stage: 5 Challenge Level:

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?

### Spring Frames

##### Stage: 5 Challenge Level:

Find the equilibrium points for these configurations of springs.

### Napoleon's Hat

##### Stage: 5 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?

### Amida

##### Stage: 5 Challenge Level:

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?

### Rain or Shine

##### Stage: 5 Challenge Level:

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.