Explaining, convincing and proving

Is it possible to arrange the numbers 1-6 on the nodes of this diagram, so that all the sums between numbers on adjacent nodes are different?

Generalise this inequality involving integrals.

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

Pizza, Indian or Chinese takeaway? If everyone liked at least one, how many only liked Indian?

Which digit replaces x in this crossnumber?

If it takes 852 digits to number all the pages of a book, what is the number of the last page?

One of N coins is slightly heavier than the others. How large can N be if the coin can be determined with only two weighings with a set of scales?

What is the largest number Sophie can use to have seven positive integers with a mean of 7?

In a supermarket, there are two lines of tightly packed trolleys. What is the length of one trolley?

Weekly Problem 41 - 2016
The diagram shows a square, with lines drawn from its centre. What is the shaded area?