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Puzzles

Two triangles

Source: Maths Jam
The three sides of this triangle have been split into three equal parts and three lines have been added.
What is the area of the smaller blue triangle as a fraction of the area of the original large triangle?

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Dodexagon

In the diagram, B, A, C, D, E, F, G, H, I, J, K and L are the vertices of a regular dodecagon and B, A, M, N, O and P are the vertices of a regular hexagon.
Show that A, M and E lie on a straight line.

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Equal side and angle

In the diagram shown, the lengths \(AD = CD\) and the angles \(ABD=CBD\).
Prove that the lengths \(AB=BC\).

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Arctan

Prove that \(\arctan(1)+\arctan(2)+\arctan(3)=\pi\).

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Square deal

This unit square is divided into four regions by a diagonal and a line that connects a vertex to the midpoint of an opposite side. What are the areas of the four regions?

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Half an equilateral triangle

Source: Maths Jam
What is the shortest straight line which bisects an equilateral triangle?

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Four points on a shape

Fiona wants to draw a 2-dimensional shape whose perimeter passes through the points A, B, C and D
Which of the following shapes can she draw?
(i) A circle
(ii) An equilateral triangle
(iii) A square

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Pizza

Twelve friends want to share a pizza. One of the friends is very fussy and will not eat the centre of the pizza.
Is it possible to split a (circular) pizza into twelve identical pieces such that there is at least one piece which does not touch the centre?

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