Problems

In a parliament with only one house every member had not more than three enemies. Is it possible split this parliament into two separate houses in such a way that each member will have not more than one enemy in the same house as them. We assume that hard feelings among members of parliament are mutual, namely if \(A\) recognises \(B\) as their enemy, then \(B\) also recognises \(A\) as their enemy.

Find all such \(n\), that the closed system of \(n\) gears on a plane can move. We call a system closed if the first gear wheel is connected to the second and the \(n\)th, the second is connected to the first and the third, and so on. On the picture we have a closed system of three gears.

Find the angles of the triangle \(ABC\) if the center of the inscribed circle \(E\) and the center of the superscribed circle \(D\) are symmetric with respect to the segment \(AC\).

Katie and Charlotte had \(4\) sheets of paper. They decided to cut some of the sheets into \(4\) pieces, then, some of the newly obtained papersheets they also cut into \(4\). In the end they counted the number of all sheets. Could this number be \(2024\)?

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 \(a_1,a_2,...a_n\).

A useful common problem-solving strategy is to divide a problem into cases. We can divide the problem into familiar and unfamiliar cases; easy and difficult cases; typical and extreme cases etc. The division is sometimes suggested by the problem, but oftentimes requires a bit of work first.

If you are stuck on a problem or you are not sure where to begin, gathering data by trying out easy or typical cases first might help you with the following (this list is not exhaustive):

1. Gaining intuition of the problem

2. Isolating the difficulties

3. Quantifying progress on the problem

4. Setting up or completing inductive arguments

Let us take a look at this strategy in action.

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 \(ARTS+STAR=10,T31\).

Split the numbers from \(1\) to \(9\) into three triplets such that the sum of the three numbers in each triplet is prime. For example, if you split them into \(124\), \(356\) and \(789\), then the triplet \(124\) is correct, since \(1+2+4=7\) is prime. But the other two triples are incorrect, since \(3+5+6=14\) and \(7+8+9=24\) are not prime.