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Square Mean

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

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Can you use the diagram to prove the AM-GM inequality?

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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?

Partial Means

Stage: 4 Short Challenge Level: Challenge Level:1

The sum of all $64$ numbers is their mean times the number of numbers, which is $64 \times 64 = 4096$. Similarly the sum of the first $36$ numbers is $36^2 = 1296$. Therefore the sum of the last $28$ numbers is $64^2-36^2= 4096 - 1296 = 2800$. Therefore the mean of the last $28$ numbers is $\frac{2800}{28}=100$.

This problem is taken from the UKMT Mathematical Challenges.
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