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?

Inside Outside

Stage: 4 Challenge Level:

A good start by Jia and Jeremy from Raffles Institution, Singapore.

To balance a bar balance, equal weights are required on each side of the pivot.

Taking $a$ as the distance of the $3$ weight from the pivot, and $b$ as the distance of the $2$ weight from the pivot. And using each column from the pivot as a distance of one, the weight on the left of the pivot has a moment of $40$.

So $3a+2b$ must equal (that is, balance) $40$.

Matching pairs of $a$ and $b$ could be:

$$a = 5 \quad\mbox{and}\quad b = 12.5$$
or
$$a = 4 \quad\mbox{and}\quad b = 14\;.$$

And some good algebra reasoning from Joan in Edinburgh.

If we want $a$ less than $b$ it would be good to find where $a = b$, that means that the two weights are together at the same place.

If $a = b$ then $a$ and $b$ have to be $8$ (from $5a = 40$ or $5b = 40$ whichever you prefer).

But we want $a$ less than $b$, so a can't be more than $8$.
The $3$ weight can go as near to the pivot as we like and $b$ will just have to get bigger to keep the balance. If the $3$ weight does go to zero $b$ will have to be $20$.

$20$ is the furthest out from the pivot that the $2$ weight goes if it keeps balanced with the $4$ weight on the other side.