At a round table, there are 10 people, each of whom is either a knight who always speaks the truth, or a liar who always lies. Two of them said: “Both my neighbors are liars,” and the remaining eight stated: “Both my neighbors are knights.” How many knights could there be among these 10 people?
There are 23 students in a class. During the year, each student of this class celebrated their birthday once, which was attended by some (at least one, but not all) of their classmates. Could it happen that every two pupils of this class met each other the same number of times at such celebrations? (It is believed that at every party every two guests met, and also the birthday person met all the guests.)
Author: G. Zhukov
The square trinomial \(f (x) = ax^2 + bx + c\) that does not have roots is such that the coefficient \(b\) is rational, and among the numbers \(c\) and \(f (c)\) there is exactly one irrational.
Can the discriminant of the trinomial \(f (x)\) be rational?
A mix of boys and girls are standing in a circle. There are 20 children in total. It is known that each boys’ neighbour in the clockwise direction is a child wearing a blue T-shirt, and that each girls’ neighbour in the anticlockwise direction is a child wearing a red T-shirt. Is it possible to uniquely determine how many boys there are in the circle?
In a \(10 \times 10\) square, all of the cells of the upper left \(5 \times 5\) square are painted black and the rest of the cells are painted white. What is the largest number of polygons that can be cut from this square (on the boundaries of the cells) so that in every polygon there would be three times as many white cells than black cells? (Polygons do not have to be equal in shape or size.)
A firm recorded its expenses in pounds for 100 items, creating a list of 100 numbers (with each number having no more than two decimal places). Each accountant took a copy of the list and found an approximate amount of expenses, acting as follows. At first, he arbitrarily chose two numbers from the list, added them, discarded the sum after the decimal point (if there was anything) and recorded the result instead of the selected two numbers. With the resulting list of 99 numbers, he did the same, and so on, until there was one whole number left in the list. It turned out that in the end all the accountants ended up with different results. What is the largest number of accountants that could work in the company?
Author: A. Khrabrov
Do there exist integers \(a\) and \(b\) such that
a) the equation \(x^2 + ax + b = 0\) does not have roots, and the equation \(\lfloor x^2\rfloor + ax + b = 0\) does have roots?
b) the equation \(x^2 + 2ax + b = 0\) does not have roots, and the equation \(\lfloor x^2\rfloor + 2ax + b = 0\) does have roots?
Note that here, square brackets represent integers and curly brackets represent non-integer values or 0.
An abstract artist took a wooden \(5\times 5\times 5\) cube and divided each face into unit squares. He painted each square in one of three colours – black, white, and red – so that there were no horizontally or vertically adjacent squares of the same colour. What is the smallest possible number of squares the artist could have painted black following this rule? Unit squares which share a side are considered adjacent both when the squares lie on the same face and when they lie on adjacent faces.
a) There is an unlimited set of cards with the words “abc”, “bca”, “cab” written. Of these, the word written is determined according to this rule. For the initial word, any card can be selected, and then on each turn to the existing word, you can either add on a card to the left or to the right, or cut the word anywhere (between the letters) and put a card there. Is it possible to make a palindrome with this method?
b) There is an unlimited set of red cards with the words “abc”, “bca”, “cab” and blue cards with the words “cba”, “acb”, “bac”. Using them, according to the same rules, a palindrome was made. Is it true that the same number of red and blue cards were used?
A cubic polynomial \(f (x)\) is given. Let’s find a group of three different numbers \((a, b, c)\) such that \(f (a)= b\), \(f (b) = c\) and \(f (c) = a\). It is known that there were eight such groups \([a_i, b_i, c_i]\), \(i = 1, 2, \dots , 8\), which contains 24 different numbers. Prove that among eight numbers of the form \(a_i + b_i + c_i\) at least three are different.