Proof - June 2005, Stage 4&5

Problems

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Pentagonal

Stage: 4 Challenge Level: Challenge Level:1

Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)? Can you prove it to be true for a rectangle or a hexagon?

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Screen Shot

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

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Two Points Plus One Line

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

Draw a line (considered endless in both directions), put a point somewhere on each side of the line. Label these points A and B. Use a geometric construction to locate a point, P, on the line,. . . .

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Golden Construction

Stage: 5 Challenge Level: Challenge Level:1

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

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Golden Eggs

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

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Golden Fractions

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.