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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}.

Quad in Quad

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

Loopy

Age 14 to 16
Challenge Level
The terms $a_1, a_2, a_3,...\ a_n,...\ $ of a sequence are given by: $$a_n =\frac{1+a_{n-1}}{a_{n-2}}.$$ Investigate the sequences you get when you choose your own first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture? Investigate the sequences.