Reasoning, convincing and proving

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?

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by 3, 5 and by 7...

In a league of 5 football teams which play in a round robin tournament show that it is possible for all five teams to be league leaders.

Can you work out the number of chairs at a cafe from the number of legs?