### Two Cubes

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

### Rationals Between...

What fractions can you find between the square roots of 65 and 67?

### Square Mean

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

# Mediant

##### Stage: 4 Challenge Level:

If you take two tests and get $a$ marks out of a maximum $b$ in the first, scoring ${a\over b}$, and you do better with $c$ marks out of $d$ in the second, scoring ${c\over d}$, you expect the result of combining the marks for the two tests, the mediant ${{a+c}\over {b+d}}$, to lie between the results for the two tests separately. If 0 < ${a\over b}$ < ${c\over d}$ prove, with reference to the diagram, that $${a\over b} < {{a+c}\over {b+d}} < {c\over d}.$$