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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?

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A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

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Gift of Gems

Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one apiece to the rest in token of regard; and they thus became owners of stock of precisely equal value. Tell me the prices of their gems and the values of their stocks. (Editors note: In this problem you are only given enough information to find the values of rubies, saphires and diamonds relative to the price of a pearl.)

Areas and Ratios

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
What happens to the area of a triangle if I keep the height the same but double the base? Or treble it? Or...?