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Polycircles
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
DOTS Division
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
Loopy
Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?
Age 14 to 16 Challenge Level:
ABCD is a convex quadrilateral. The points P, Q, R and S are the midpoints of the edges of ABCD. You can change the shape of the quadrilateral ABCD.
What do you notice about the quadrilateral PQRS as ABCD changes?
Is the area of PQRS always the same fraction of the area of ABCD and, if so, what is this fraction?
Try to prove your conjectures.
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NRICH is part of the family of activities in the Millennium Mathematics Project.