Problems

Prove the triangle inequality: in any triangle \(ABC\) the side \(AB < AC+ BC\).

In certain kingdom there are a lot of cities, it is known that all the distances between the cities are distinct. One morning one plane flew out of each city to the nearest city. Could it happen that in one city landed more than \(5\) planes?

On a \(10\times 10\) board, a bacterium sits in one of the cells. In one move, the bacterium shifts to a cell adjacent to the side (i.e. not diagonal) and divides into two bacteria (both remain in the same new cell). Then, again, one of the bacteria sitting on the board shifts to a new adjacent cell, either horizontally or vertically, and divides into two, and so on. Is it possible for there to be an equal number of bacteria in all cells after several such moves?

Prove that the set of all finite subsets of natural numbers \(\mathbb{N}\) is countable. Then prove that the set of all subsets of natural numbers is not countable.

In a scout group among any four participants there is at least one, who knows three other. Prove that there is at least one participant, who knows the rest of the group.

The distance between two villages equals \(999\) kilometres. When you go from one village to the other, every kilometre you see signs along the road, saying \(0 \mid 999, \, 1\mid 998, \, 2\mid 997, ..., 999\mid 0\). Find the number of signs, that contain only two different digits.

The first player is thinking about a finite sequence of numbers \(a_1,a_2, ..., a_n\). The second player can try to find the sequence by naming his own sequence \(b_1, b_2, ...b_n\), after that the first player will tell the result \(a_1b_1 + a_2b_2 + ...a_nb_n\). In the next step the second player can say another sequence \(c_1, c_2, ...c_n\) to get another answer \(a_1c_1+ a_2c_2 + ... a_nc_n\). Find the smallest amount of steps the second player has to take to find out the sequence \(a_1,a_2,...a_n\).

Michael made a cube with edge \(1\) out of eight bars as on the picture. It is known that all the bars, regardless of color have the same volume, the grey bars are the same and the white bars are also the same. Find the lengths of the edges of the white bar.

Red, blue and green chameleons live on the island, one day \(35\) chameleons stood in a circle. A minute later, they all changed color at the same time, each changed into the color of one of their neighbours. A minute later, everyone again changed the colors at the same time into the color of one of their neighbours. Could it turn out that each chameleon turned red, blue, and green at some point?

Is it possible to paint \(15\) segments in the picture below in three colours in such a way, that no three segments of the same colour have a common end?