Problems

Father Christmas has an infinite number of sweets. A minute before the New Year, Father Christmas gives some children 100 sweets, while the Snow Maiden takes one sweet from them. Within half a minute before the New Year, Father Christmas gives the children 100 more sweets, and the Snow Maiden again takes one sweet. The same is repeated for 15 seconds, for 7.5 seconds, etc. until the new Year. Prove that the Snow Maiden will be able to take away all the sweets from the children by the New Year.

Let \(x\) be a natural number. Among the statements:

\(2x\) is more than 70;

\(x\) is less than 100;

\(3x\) is greater than 25;

\(x\) is not less than 10;

\(x\) is greater than 5;

three are true and two are false. What is \(x\)?

A square \(4 \times 4\) is called magic if all the numbers from 1 to 16 can be written into its cells in such a way that the sums of numbers in columns, rows and two diagonals are equal to each other. Sixth-grader Edwin began to make a magic square and written the number 1 in certain cell. His younger brother Theo decided to help him and put the numbers \(2\) and \(3\) in the cells adjacent to the number \(1\). Is it possible for Edwin to finish the magic square after such help?

The king possesses \(7\) bags of gold coins, each containing \(100\) coins. While the coins in each bag appear identical, they vary in weight and they cannot be told apart by looking. The king recalls that within these bags, one contains coins that weigh \(7\)g each, another has coins weighing \(8\)g, the third bag contains coins weighing \(9\)g, the fourth has coins weighing \(10\)g, the fifth contains coins weighing \(11\)g, the sixth holds coins weighing \(12\)g, and finally, the seventh bag contains coins weighing \(13\)g each. However, he cannot remember which bag corresponds to which coin weight.
The king reported his situation to his chancellor, pointing to one of the bags, and asked how to determine the weight of the coins in that bag. The chancellor has large two-cup scales without weights. These scales can precisely indicate whether the weights on the cups are equal or, if not, which cup is heavier. Can the chancellor ascertain which coins are in the bag indicated by the king, using no more than two weightings? The chancellor is permitted to take as many coins as necessary to conduct the weightings.

There are \(20\) chairs in the room, which come in two colors: blue and red. Each chair is occupied by either a knight or a liar. Knights always tell the truth, while liars always lie. Initially, each of those seated claimed to be sitting on a blue chair. Then, they switched seats, after which half of the participants asserted that they were now sitting on blue chairs, while the other half claimed to be sitting on red ones. How many knights are currently occupying red chairs?

A labyrinth was drawn on a \(5\times 5\) grid square with an outer wall and an exit one cell wide, as well as with inner walls running along the grid lines. In the picture, we have hidden all the inner walls from you (We give you several copies to facilitate drawing) 
Please draw how the walls were arranged. Keep in mind that the numbers in the cells represent the smallest number of steps needed to exit the maze, starting from that cell. A step can be taken to any adjacent cell vertically or horizontally, but not diagonally (and only if there is no wall between them, of course).

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?

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?