The positive real numbers \(a, b, c, x, y\) satisfy the following system of equations: \[\left\{ \begin{aligned} x^2 + xy + y^2 = a^2\\ y^2 + yz + z^2 = b^2\\ x^2 + xz + z^2 = c^2 \end{aligned} \right.\]
Find the value of \(xy + yz + xz\) in terms of \(a, b,\) and \(c.\)
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 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: \[\left\{ \begin{aligned} (x+y)^3=z\\ (x+z)^3=y\\ (y+z)^3=x \end{aligned} \right.\]
In the first room, there are two doors. The signs on them say:
There is treasure behind this door, and a trap behind the other door.
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:
There is a treasure behind both doors.
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?