### Square Mean

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

### AMGM

Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

### Cubic Rotations

There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube?

# Weekly Problem 12 - 2006

##### Stage: 4 Challenge Level:

$34 \frac{2}{7}$

Let the distance between Newcastle and South Shields be $x$ miles. Then the time for the whole journey is $$\left(\frac{x}{30}+ \frac{x}{40}\right) \; \text{hours} = \frac{7x}{120}\; \text{hours}$$ Thus the average speed is $$\left(2x \div \frac{7x}{120}\right)\; \text{mph} = \frac{240}{7} \; \text{mph} = 34\frac{2}{7} \; \text{mph}$$

This problem is taken from the UKMT Mathematical Challenges.

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