Reasoning, convincing and proving

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

An inequality involving integrals of squares of functions.

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

Jack does a 20-question quiz. How many questions didn't he attempt?

Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?

You may have met Magic Squares, now meet an Anti-Magic Square. Its properties are slightly different - can you still solve it?

Can the pdfs and cdfs of an exponential distribution intersect?

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Can you find different ways of creating paths using these paving slabs?