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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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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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At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Out of the Window

Stage: 4 Short Challenge Level: Challenge Level:1

By Pythagoras Theorem, the diagonal of the window is 100 cm, which exceeds the length or breadth of all the sheets. So the first three pieces can go through the window either way and the 90 x 105 cm piece can also go through the window, provided the 90 cm edge goes first.

This problem is taken from the UKMT Mathematical Challenges.