Problems

Let \(a\) and \(b\) be real numbers. Find a representation of \(a^3 + b^3\) as a product.

• Find a representation of the number \(117 = 121-4\) as a product.

• Let \(a\) and \(b\) be real numbers. Find a representation of \(a^2 - b^2\) as a product.

Solve the system of equations in real numbers: \[\left\{ \begin{aligned} x+y = 2\\ xy-z^2 = 1 \end{aligned} \right.\]

Find all solutions of the equation: \(xy + 1 = x + y\).

Find all solutions of the system of equations: \[\left\{ \begin{aligned} x+y+z = a\\ x^2 + y^2+z^2 = a^2\\ x^3+y^3+z^3 = a^3 \end{aligned} \right.\]

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?