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Puzzles

22 December

There are 4 ways to pick three vertices of a regular quadrilateral so that they form a right-angled triangle:
In another regular polygon with \(n\) sides, there are 14620 ways to pick three vertices so that they form a right-angled triangle. What is \(n\)?

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20 December

The diagram to the right shows (two copies of) quadrilateral ABCD.
The sum of the angles ABC and BCD (green and blue in quadrilateral on the left) is 180°. The sum of the angles ABC and DAB (green and orange in quadrilateral on the left) is also 180°. In the diagram on the right, a point inside the quadrilateral has been used to draw two triangles.
The area of the quadrilateral is 850. What is the smallest that the total area of the two triangles could be?

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7 December

What is the area of the largest triangle that fits inside a regular hexagon with area 952?

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20 December

What is the area of the largest area triangle that has one side of length 32 and one side of length 19?

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13 December

The diagram to the left shows three circles and two triangles. The three circles all meet at one point. The vertices of the smaller red triangle are at the centres of the circles. The lines connecting the vertices of the larger blue triangle to the point where all three circles meet are diameters of the three circles.
The area of the smaller red triangle is 226. What is the area of the larger blue triangle?

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7 December

The picture below shows eight regular decagons. In each decagon, a red triangle has been drawn with vertices at three of the vertices of the decagon.
The area of each decagon is 240. What is the total area of all the red triangles?

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5 December

How many different isosceles triangles are there whose perimeter is 50 units, and whose area is an integer number of square-units?
(Two triangles that are rotations, reflections and translations of each other are counted as the same triangle. Triangles with an area of 0 should not be counted.)

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19 December

The diagram to the right shows a triangle. Two of the sides of the triangle have been split into three pieces, with lines drawn from the opposite vertex. In total, the diagram now contains 27 triangles of any size.
Another triangle has two of its sides split into eight pieces, with lines drawn from the opposite vertex. How many triangles (of any size) would this create?

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