Find the coefficient of \(x\) for the polynomial \((x - a) (x - b) (x - c) \dots (x - z)\).
To a certain number, we add the sum of its digits and the answer we get is 2014. Give an example of such a number.
An incredible legend says that one day Stirling was considering the numbers of Stirling of the second kind. During his thoughtfulness, he threw 10 regular dice on the table. After the next throw, he suddenly noticed that in the dropped combination of points there were all of the numbers from 1 to 6. Immediately Stirling reflected: what is the probability of such an event? What is the probability that when throwing 10 dice each number of points from 1 to 6 will drop out on at least one die?
There is a system of equations \[\begin{aligned} * x + * y + * z &= 0,\\ * x + * y + * z &= 0,\\ * x + * y + * z &= 0. \end{aligned}\] Two people alternately enter a number instead of a star. Prove that the player that goes first can always ensure that the system has a non-zero solution.
A White Rook pursues a black bishop on a board of \(3 \times 1969\) cells (they walk in turn according to the usual rules). How should the rook play to take the bishop? White makes the first move.
The White Rook pursues a black horse on a board of \(3 \times 1969\) cells (they walk in turn according to the usual rules). How should the rook play in order to take the horse? White makes the first move.
On an island there are 1,234 residents, each of whom is either a knight (who always tells the truth) or a liar (who always lies). One day, all of the inhabitants of the island were broken up into pairs, and each one said: “He is a knight!" or “He is a liar!" about his partner. Could it eventually turn out to be that the number of “He is a knight!" and “He is a liar!" phrases is the same?
Solving the problem: “What is the solution of the expression \(x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^3 + 2000x^2 + 2000x + 3000\) (\(x\) is a real number) if \(x^2 + x + 1 = 0\)?”, Vasya got the answer of 3000. Is Vasya right?
The best student in the class, Katie, and the second-best, Mike, tried to find the minimum 5-digit number which consists of different even numbers. Katie found her number correctly, but Mike was mistaken. However, it turned out that the difference between Katie and Mike’s numbers was less than 100. What are Katie and Mike’s numbers?
The old shoemaker Carl sewed some boots and sent his son Hans to the market to sell them for £25. Two disabled people came to the boy’s market stall (one without a left leg, the other without a right one) and was asked to sell each of them a boot. Hans agreed and sold each boot for £12.50.
When the boy came home and told the whole story to his father, Carl decided that his son should have sold the boots to the disabled buyers for less – each for £10. He gave Hans £5 and ordered him to return £2.50 to each disabled buyer.
While the boy was looking for the disabled people at the market, he saw that someone was selling sweets and as could not resist, spent £3 on sweets. After that, he found the disabled buyers and gave them the remaining money – each got £1. Returning home, Hans realised how badly he had acted. He told his father and asked for forgiveness. The shoemaker was very angry and punished his son by sending him to his room.
Sitting in his room, Hans thought about the day’s events. It turned out that since he returned £1 to each buyer, they paid £11.50 for each boot: \(12.50 - 1 = 11.50\). So, the boots cost £23: \(2 \times 11.50 = 23\). And Hans spent £3 on sweets, therefore, it total, there were £26: \(23 + 3 = 26\). But there were only £25! Where did the extra pound come from?