### 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?

# Tyneside Average Speed

##### Stage: 4 Short Challenge Level:

A van travels from Newcastle to South Shields at an average speed of $30$ mph and returns by the same route at an average speed of $40$ mph. What is the van's average speed for the whole journey?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.
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