Explaining, convincing and proving

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?

Each digit of a positive integer is 1, 2 or 3, and each of these occurs at least twice. What is the smallest such integer that is not divisible by 2 or 3?

Weekly Problem 31 - 2017
The triangle HIJ has the same area as the square FGHI. What is the distance from J to the line extended through F and G?

Weekly Problem 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?

Can the pdfs and cdfs of an exponential distribution intersect?

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Can you explain why a sequence of operations always gives you perfect squares?

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

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