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.

Fibonacci Factors

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

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?

Trig Rules OK

Age 16 to 18 Challenge Level:

Draw any two squares which meet at a common vertex $C$ and join the adjacent vertices to make two triangles $CAB$ and $CDE$.

Construct the perpendicular from $C$ to $AB$, (the altitude of the triangle). When you extend this line where does it cut $DE$?

Now bisect the line $AB$ to find the midpoint of this line $M$. Draw the median $MC$ of triangle $ABC$ and extend it to cut $DE$. What do you notice about the lines $MC$ and $DE$?

Will you get the same results about the two triangles formed if you draw squares of different sizes or at different angles to each other? Make a conjecture about the altitude of one of these triangles and prove your conjecture.

Thank you Geoff Faux for suggesting this problem.