Problems

Find all solutions of the system of equations: \[\left\{ \begin{aligned} (x+y)^3=z\\ (x+z)^3=y\\ (y+z)^3=x \end{aligned} \right.\]

Sometimes different areas in mathematics are more related than they seem to be. A lot of algebraic expressions have geometric interpretation, and a lot of them can be used to solve problems in number theory.

Today we will solve several logic problems that revolve about a very simple idea. Imagine you are in a room in a dungeon and you can see doors leading out of the room. Some of them lead to the treasure and some of them lead to traps. It is possible that all doors lead to treasure or all lead to traps, but it is also possible that one door leads to treasure and all other lead to traps. Unless specified, there is always something behind the door.
Each door has a sign with a statement on it, but those statements are not always true. You have a dungeon guide, who is always honest with you and will tell you something about the truthfulness of the statements on the doors, but it will be up to you to put it all together and pick the correct door... or walk away, if you believe there is no treasure.

In the first room, there are two doors. The signs on them say:

1. There is treasure behind this door, and a trap behind the other door.

2. Behind one of these doors there is treasure and behind the other there is a trap.

Your guide says: One of the signs is true and the other is false. Which door will you open?

In the second room, there are two doors. Both statements on them say:

1. There is a treasure behind both doors.

2. There is a treasure behind both doors.

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?

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?