Reasoning, convincing and proving

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Can you use the diagram to prove the AM-GM inequality?

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try to explain why this works.

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Follow the hints and prove Pick's Theorem.

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?