The World's Hardest Maths Problems

Over the years, in many forms or another, people have been set Maths problems, either by other people or by God. Some of these problems however are infinitely harder than the rest, and The World's Hardest Maths Problems is a collection of these for your enjoyment, or rather the lack of it.

History

These individual questions were first created by Euclidean Protectorate in order to pose a test for the officers to see if they were fit to join the ranks in the fight against the Non-Euclidean Witchcraft created during World War π. The idea was that there could only be one of three outcomes, each with different decisions made concerning the individual to have provided those answers:

• The officer attempted to answer the questions in the time allowed, and failed on account of their impossibility. If they did not complain, and merely took it on themselves as failure, then they were counted as having enough faith in the Order to be allowed to fight.

• However, if the officer complained that the questions were impossible, then whilst their faith was still demonstrated, they were shot for insubordination. In the order there can be no room for such insolence!

• The third outcome was that the officer would answer all the questions, and worst of all get them right. As this was only possible by practicing Non-Euclidean Magicks, the officer has deemed a Heretic and burned at the stake for witchcraft

Objections!

Many (i.e. most) historians argue that the above section is utter nonsense, particularly as the whole World War π thing never happened anyway. Instead they hold that the questions are merely the product of a deranged mind that has been spending too much time with Puff the Magic Dragon, if you catch my meaning.

The Questions

Information required to complete the questions

You are given a triangle that has sides of 66cm, 73cm, and 94cm. One of the angles is right-angled (meaning that it is possible by trial and error to calculate what each of the angles are). Inside this triangle is a square, so that three corners are in contact with the lines bounding the triangle. One of the sides or the square, which we shall now dub z, is also tangent to a circle, with a radius such that the centre of the circle lies along the side of the triangle with length 73cm. You are also given a regular octagon, which you are told is the same area as the total are of the circle and triangle if they are taken together (i.e. the overlapping area is not counted twice), and one side of this octagon forms another side of equal length belonging to a second square. The area of this square is dubbed x.

Question 1

Give the value, to three significant figures, of x.

[15 marks]

Question 2

An isosceles triangle is drawn so that it has the same area as the above square (i.e. x), and with two sides that are equal to the square root of x (henceforth dubbed y). What is the length of the third side?

[100+90-7685*58/92 marks]

Question 3

Prove that the triangle above exists.

[25 marks]

Question 4

What is the area of a octagon of side length y, in cubic inches. (Note that this question uses non-euclidean goemetry)

[2πr marks]

Question 5

Through cunning use of Pythaogoras' Theorem, prove that aliens do not exist.

[-0 marks]

Question 6

If $48y^{2}+{\pi }y-6778=x$, then what does y smell like?

[-10 marks]