Problems

Find all solutions of the equation: \(x^2 + y^2 + z^2 + t^2 = x(y + z + t)\).

Let \(a\) and \(b\) be real numbers. Find a way to write \(a^3 + b^3\) as a product of terms involving \(a\) and \(b\).

• 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: \[\left\{ \begin{aligned} (x+y)^3=z\\ (x+z)^3=y\\ (y+z)^3=x \end{aligned} \right.\]

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?