a) We are given two cogs, each with 14 teeth. They are placed on top of one another, so that their teeth are in line with one another and their projection looks like a single cog. After this 4 teeth are removed from each cog, the same 4 teeth on each one. Is it always then possible to rotate one of the cogs with respect to the other so that the projection of the two partially toothless cogs appears as a single complete cog? The cogs can be rotated in the same plane, but cannot be flipped over.
b) The same question, but this time two cogs of 13 teeth each from which 4 are again removed?
What is the minimum number of squares that need to be marked on a chessboard, so that:
1) There are no horizontally, vertically, or diagonally adjacent marked squares.
2) Adding any single new marked square breaks rule 1.
A staircase has 100 steps. Vivian wants to go down the stairs, starting from the top, and she can only do so by jumping down and then up, down and then up, and so on. The jumps can be of three types – six steps (jumping over five to land on the sixth), seven steps or eight steps. Note that Vivian does not jump onto the same step twice. Will she be able to go down the stairs?
Find the number of solutions in natural numbers of the equation \(\lfloor x / 10\rfloor = \lfloor x / 11\rfloor + 1\).
A gang contains 101 gangsters. The whole gang has never taken part in a raid together, but every possible pair of gangsters have taken part in a raid together exactly once. Prove that one of the gangsters has taken part in no less than 11 different raids.
An after school club was attended by 60 pupils. It turns out that in any group of 10 there will always be 3 classmates. Prove that within the group of 60 who attended there will always be at least 15 pupils from the same class.
4 points \(a, b, c, d\) lie on the segment \([0, 1]\) of the number line. Prove that there will be a point \(x\), lying in the segment \([0, 1]\), that satisfies \[\frac{1}{ | x-a |}+\frac{1}{ | x-b |}+\frac{1}{ | x-c |}+\frac{1}{ | x-d |} < 40.\]
Some real numbers \(a_1, a_2, a_3,\dots ,a _{2022}\) are written in a row. Prove that it is possible to pick one or several adjacent numbers, so that their sum is less than 0.001 away from a whole number.
17 squares are marked on an \(8\times 8\) chessboard. In chess a knight can move horizontally or vertically, one space then two or two spaces then one – eg: two down and one across, or one down and two across. Prove that it is always possible to pick two of these squares so that a knight would need no less than three moves to get from one to the other.
A square is cut by 18 straight lines, 9 of which are parallel to one side of the square and the other 9 parallel to the other – perpendicular to the first 9 – dividing the square into 100 rectangles. It turns out that exactly 9 of these rectangles are squares. Prove that among these 9 squares there will be two that are identical.