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Poly Fibs

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

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Fibonacci Factors

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

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Fixing It

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

Cyclic Triangles

Age 16 to 18 Challenge Level:

Triangle ABC inscribed in circle with diameter AB
A triangle $ABC$ is inscribed in a circle with $AB$ as diameter. Find the maximum value of $AC + CB$.

Now generalise your result to the case where $AB$ is fixed but not a diameter of the circle.

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius $r$ that has the maximum perimeter and the maximum area.