Sine, cosine, tangent

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Octa-Flower

Age

16 to 18

Challenge level

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?
problem
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Sine and Cosine

Age

14 to 16

Challenge level

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?
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Shape and Territory

Age

16 to 18

Challenge level

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
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Figure of Eight

Age

14 to 16

Challenge level

On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?
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Spokes

Age

16 to 18

Challenge level

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.
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Where Is the Dot?

Age

14 to 16

Challenge level

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?
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Doesn't Add Up

Age

14 to 16

Challenge level

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
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Inscribed in a Circle

Age

14 to 16

Challenge level

The area of a square inscribed in a circle with a unit radius is 2. What is the area of these other regular polygons inscribed in a circle with a unit radius?
problem
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Cosines Rule

Age

14 to 16

Challenge level

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.
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Far Horizon

Age

14 to 16

Challenge level

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?