Frodo can meet either Sam, or Pippin, or Merry in the fog. One day everyone came out to meet Frodo, but the fog was thick, and Frodo could not see where everyone was, so he asked each of his friends to introduce themselves.
The one who from Frodo’s perspective was on the left, said: "Merry is next to me."
The one on Frodo’s right said: "The person who just spoke is Pippin."
Finally, the one in the center announced, "On my left is Sam."
Identify who stood where, knowing that Sam always lies, Pippin sometimes lies, and Merry never lies?
Using areas of squares and rectangles, show that for any positive real numbers \(a\) and \(b\), \((a+b)^2 = a^2+2ab+b^2\).
The identity above is true for any real numbers, not necessarily positive, in fact in order to prove it the usual way one only needs to remember that multiplication is commutative and the distributive property of addition and multiplication:
\(a\times b = b\times a\);
\((a+b)\times c = a\times c + b\times c\).
Annie found a prime number \(p\) to which you can add \(4\) to make it a perfect square. What is the value of \(p\)?
Let \(a\) and \(b\) be positive real numbers. Using areas of rectangles and squares, show that \(a^2 - b^2 = (a-b) \times (a+b)\).
Try to prove it in two ways, one geometric and one algebraic.
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: \(xy + 1 = x + y\).
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?
In the third room, there are three doors. The statements on them say:
Behind this door there is a trap.
Behind this door there is treasure.
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:
There is treasure behind at least one of the doors.
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?