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Building Tetrahedra

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.

Pythagorean Quadruple

Stage: 4 Short Challenge Level: Challenge Level:1

It is clear that each of a, b and c must be less than or equal to $10$. A brief inspection will show that the only combination of different square numbers which total $121$ is $81, 36, 4$, so the $a, b$ and $c$ are $2, 6$ and $9$.

This problem is taken from the UKMT Mathematical Challenges.