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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?

Knights and Knaves

Stage: 3 and 4 Short Challenge Level: Challenge Level:1
The first person cannot be telling the truth since if all the others are knaves, this contradicts that they are telling the truth when they say the person in front is a knave.
The second person says the first is a knave so is telling the truth; he is a knight.
The third says this knight is a knave so he is lying; he is a knave. Continuing in this way we see that there is an alternating sequence of 13 knaves and 12 knights.

This problem is taken from the UKMT Mathematical Challenges.

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