Problems

In the US, it is customary to record the date as follows: the number of the month, then the number of the day and then the year. In Europe, the number comes first, then the month and then the year. How many days are there in the year, the date of which can be read definitively, without knowing how it was written?

A pedestrian walked along six streets of one city, passing each street exactly twice, but could not get around them, having passed each one only once. Could this be?

a) In how many ways can Dima paint five Christmas trees in silver, green and blue colours, if the amount of paint is unlimited, and he paints each tree in only one colour?

b) Dima has five baubles: a red, a green, a yellow, a blue and a gold one. In how many ways can he decorate five Christmas trees with them, if he needs to put exactly one bauble on each tree?

c) What about if he can hang several baubles on one Christmas tree (and all of the baubles have to be used)?

Decipher the quote from "Alice in Wonderland" from the following matrix:
\[\begin{array}{@{}*{26}{c}@{}} Y&q&o&l&u&e&c&d&a&i&n \\ w&a&r&l&a&w&e&a&t&y&k \\ s&n&t&c&a&e&k&c&e&a&m \\ t&o&d&r&w&e&a&t&a&h&r \\ a&c&n&t&n&e&o&d&t&r&h \\ n&i&d&n&l&g&m&e&x&s&z \end{array}\]

Due to a mistake in the bakery, a cake that was supposed to be shaped as two concentric pieces (like on the right diagram below) came out like the left diagram below. Find the smallest number of pieces the cake should be cut into in order to rearrange the pieces into the cake on the right side of the picture.

Note that the cake is \(\textit{not}\) tiered like a wedding cake, but is shaped like a cylinder with a flat top. Curved cuts are allowed.

The letters \(A\), \(R\), \(S\) and \(T\) represent different digits from \(1\) to \(9\). The same letters correspond to the same digits, while different letters correspond to different digits.
Find \(ART\), given that \(ARTS+STAR=10,T31\).

One cell was cut out of a \(3\times6\) rectangle, as seen in the diagram. How should you glue this cell in a different place to get a figure that can be cut into two identical ones? If needed, the resulting parts can be rotated and reflected.

Suppose you have a coffee mug made of stretchy and expandable material. How do you mold it into a donut that has a hole inside?

Label the vertices of a cube with the numbers \(1,2,3,\dots,8\) so that the sum of the labels of the four vertices of each of the six faces is the same.

Suppose you meet a person inhabiting this planet and they ask you “Am I a Goop?" What would you conclude?