Conjecturing and generalising

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

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

Find the relationship between the locations of points of inflection, maxima and minima of functions.

Watch the video to see how to add together an arithmetic sequence of numbers efficiently.

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?