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.
We are given 101 rectangles with integer-length sides that do not exceed 100.
Prove that amongst them there will be three rectangles \(A, B, C\), which will fit completely inside one another so that \(A \subset B \subset C\).
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?
Solve the equation \(x + \frac{1}{(y + 1/z)}= 10/7\) in natural numbers.
Find the number of solutions in natural numbers of the equation \(\lfloor x / 10\rfloor = \lfloor x / 11\rfloor + 1\).
10 friends sent one another greetings cards; each sent 5 cards. Prove that there will be two friends who sent cards to one another.
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.
A numerical sequence is defined by the following conditions: \[a_1 = 1, \quad a_{n+1} = a_n + \lfloor \sqrt{a_n}\rfloor .\]
Prove that among the terms of this sequence there are an infinite number of complete squares.
The function \(f(x)\) on the interval \([a, b]\) is equal to the maximum of several functions of the form \(y = C \times 10^{- | x-d |}\) (where \(d\) and \(C\) are different, and all \(C\) are positive). It is given that \(f (a) = f (b)\). Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.
Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?