Creating and manipulating expressions and formulae

Can you find the lap times of the two cyclists travelling at constant speeds?

Find b where 3723(base 10) = 123(base b).

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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.

Show that all pentagonal numbers are one third of a triangular number.

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

From this sum of powers, can you find the sum of the indices?