Alex writes natural numbers in a row: \(123456789101112...\) Counting from the beginning, in what places do the digits \(555\) first appear? For example, \(101\) first appears in the 10th, 11th and 12th places.
Frodo ventures into a thick fog where he is to meet his three companions: Sam, Merry, and Pippin. He can tell they are standing in a row, in front of him – one on the left, one in the middle, and one on the right – but he cannot see who is who. To help, he asks each of them to speak.
Remember: Sam always lies, Merry always tells the truth, and Pippin sometimes lies and sometimes tells the truth.
Here is what Frodo hears:
The one on Frodo’s left says: “Merry is next to me.”
The one on Frodo’s right says: “The person who just spoke is Pippin.”
The one in the middle says: “On my left is Sam.”
Can you work out who is standing on Frodo’s left, in the middle, and on the right?
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?