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Can you make a tetrahedron whose faces all have the same perimeter?

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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Repeat Product

Stage: 4 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Notice that
$$RRR=111 \times R$$
$$PQPQ=1000 \times P+100 \times Q +10\times P +Q=101 \times (10\times P+Q)$$
so $PQPQ\times RRR=111 \times 101 \times R\times(10 \times P+Q)$.

And we have $639027=PQPQ\times RRR=111 \times 101 \times R\times(10 \times P+Q)$ so we can divide both sides by $111\times101$ to give $$57=R\times(10\times P+Q)\;.$$

The only factors of $57$ are $1,3,19$ and $57$.

$R$ must divide $57$ and because $R$ must be a single digit number it can either be $1$ or $3$.

If $R$ is $3$ then $10\times P+Q=19$ so $P=1$ and $Q=9$.

If $R$ is $1$ then $10\times P+Q=57$ so $P=5$ and $Q=7$.

So there are two solutions (check that these both work) and in both cases $P+Q+R=13$.

This problem is taken from the UKMT Mathematical Challenges.