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Consecutive Numbers

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Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

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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?

Outside the Boxes

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

At each vertex of the triangle, four angles meet which must add up to 360°. Therefore altogether the twelve angles add up to 3 x 360°. Six of the angles are right-angles, and three of them are the internal angles of the triangle, which add up to 180°.

So the sum of the remaining three angles must be (3 x 360°) - (6 x 90°) - 180° = 360°.

This problem is taken from the UKMT Mathematical Challenges.

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