Problems
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 \(a_1,a_2,...a_n\).
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.
One cell was cut out of a rectangle \(3\times 6\). How to glue this cell in another place to get a figure that can be cut into two identical ones? The resulting parts can be rotated and reflected.
In the arithmetic puzzle different letters denote different digits and the same letters denote the same digit. \[P.Z + T.C + D.R + O.B + E.Y\] It turned out that all five terms are not integers, but the sum itself is an integer. Find the sum of the expression. For each possible answer, write one example with these five terms. Explain why other numbers cannot be obtained.
