Problems

In the third room, there are three doors. The statements on them say:

1. Behind this door there is a trap.

2. Behind this door there is treasure.

3. There is a trap behind the second door.

Your guide says: There is treasure behind one of the doors exactly. At most one of the three signs is true - but it is possible all of them are false.
Which door will you open?

There are two doors in the room with the following signs:

1. There is treasure behind at least one of the doors.

2. There is a trap behind the first door.

Your guide says: The signs are either both true or both false.
Which door will you open?

There are three doors with the following statements:

1. Behind the second door there is a trap.

2. Behind this door there is a trap.

3. A trap is behind the first door.

Your guide says: There is treasure behind one of the doors exactly. The sign on that door is true, but at least one of the other ones will be false.
Which door will you open?

There are two doors with the following signs:

1. There is either a trap behind this door or there is treasure behind the second door.

2. There is treasure behind the first door.

Your guide says: The signs are either both true or both false. Which door will you open?

Now you have two doors with the statements:

1. It makes no difference which door you pick.

2. There is treasure behind the first door.

Your guide says: The first sign is true if there is treasure behind the first door, otherwise it is false. The second sign is false if there is treasure behind the second door, otherwise it is true. What do you do?

Now there are three doors with statements on them:

1. There is nothing behind the third door.

2. There is a trap behind the first door.

3. There is nothing behind this door.

Your guide says: There is treasure behind one of the doors, trap behind another one and there is nothing behind the third door. The sign on the door leading to treasure is true, the sign on the door leading to a trap is false, and the third sign might be true or false.
Which door will you open, if you really really want the treasure?

This is a famous problem, called Monty Hall problem after a popular TV show in America.
In the problem, you are on a game show, being asked to choose between three doors. Behind each door, there is either a car or a goat. You choose a door. The host, Monty Hall, picks one of the other doors, which he knows has a goat behind it, and opens it, showing you the goat. (You know, by the rules of the game, that Monty will always reveal a goat.) Monty then asks whether you would like to switch your choice of door to the other remaining door. Assuming you prefer having a car more than having a goat, do you choose to switch or not to switch?

Find a representation as a product of \(a^{2n+1} + b^{2n+1}\) for general \(a,b,n\).

Find a representation as a product of \(a^n - b^n\) for general \(a,b,n\).

Let \(a,b,c,d\) be positive real numbers. Prove that \((a+b)\times(c+d) = ac+ad+bc+bd\). Find both algebraic solution and geometric interpretation.