Reasoning, convincing and proving

Show that it is rare for a ratio of ratios to be rational.

Follow the hints and prove Pick's Theorem.

In this addition each letter stands for a different digit, with S standing for 3. What is the value of YxO?

Who is the youngest in this family?

The 30 students in a class have 25 different birthdays between them. What is the largest number that can share any birthday?

Try grouping the dominoes in the ways described. Are there any left over each time? Can you explain why?

An inequality involving integrals of squares of functions.

Generalise this inequality involving integrals.

The clues for this Sudoku are the product of the numbers in adjacent squares.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?