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The function F is given on the whole real axis, and for each x the equality holds: F $(x + 1)$ F $(x)$ + F $(x + 1)$ + 1 = 0.

Prove that the function F can not be continuous.

In the dense dark forest ten sources of dead water are erupting from the ground: named from N 1 to N 10. Of the first nine sources, dead water can be taken by everyone, but the source N 10 is in the cave of the dark wizard, from which no one, except for the dark wizard himself, can collect water. The taste and color of dead water is no different from ordinary water, however, if a person drinks from one of the sources, then he will die. Only one thing can save him: if he then drinks poison from a source whose number is greater. For example, if he drinks from the seventh source, then he must necessarily drink poison from the N 8, N 9 or N 10 sources. If he doesn’t drink poison from the seventh source, but does from the ninth, only the poison from the source N 10 will save him. And if he originally drinks the tenth poison, then nothing will help him now. Robin Hood summoned the dark wizard to a duel. The terms of the duel were as follows: each brings with him a mug of liquid and gives it to his opponent. The dark wizard was delighted: “Hurray, I will give him poison No. 10, and Robin Hood can not be saved!” And I’ll drink the poison, which Robin Hood brings to me, then ill drink the N10 poison and that will save me! ” On the appointed day, both opponents met at the agreed place. They honestly exchanged mugs and drank what was in them. However, afterwards erupted the joy and surprise of the inhabitants of the dark forest, when it turned out that the dark wizard had died, and Robin Hood remained alive! Only the Wise Owl was able to guess how Robin Hood had managed to defeat dark wizard. Try and guess as well.

The judges of an Olympiad decided to denote each participant with a natural number in such a way that it would be possible to unambiguously reconstruct the number of points received by each participant in each task, and that from each two participants the one with the greater number would be the participant which received a higher score. Help the judges solve this problem!

The tracks in a zoo form an equilateral triangle, in which the middle lines are drawn. A monkey ran away from its cage. Two guards try to catch the monkey. Will they be able to catch the monkey if all three of them can run only along the tracks, and the speed of the monkey and the speed of the guards are equal and they can always see each other?

Two players play on a square field of size 99 × 99, which has been split onto cells of size 1 × 1. The first player places a cross on the center of the field; After this, the second player can place a zero on any of the eight cells surrounding the cross of the first player. After that, the first puts a cross onto any cell of the field next to one of those already occupied, etc. The first player wins if he can put a cross on any corner cell. Prove that with any strategy of the second player the first can always win.

There was a football match of 10 versus 10 players between a team of liars $($ who always lie $)$ and a team of truth-tellers $($ who always tell the truth $)$. After the match, each player was asked: “How many goals did you score?” Some participants answered “one”, Callum said “two”, some answered “three”, and the rest said “five”. Is Callum lying if it is known that the truth-tellers won with a score of 20: 17?

Four outwardly identical coins weigh 1, 2, 3 and 4 grams respectively.

Is it possible to find out in four weighings on a set of scales without weights, which one weighs how much?

On the school board a chairman is chosen. There are four candidates: A, B, C and D. A special procedure is proposed – each member of the council writes down on a special sheet of candidates the order of his preferences. For example, the sequence ACDB means that the councilor puts A in the first place, does not object very much to C, and believes that he is better than D, but least of all would like to see B. Being placed in first place gives the candidate 3 points, the second – 2 points, the third – 1 point, and the fourth – 0 points. After collecting all the sheets, the election commission summarizes the points for each candidate. The winner is the one who has the most points.

After the vote, C $($who scored fewer points than everyone $)$ withdrew his candidacy in connection with his transition to another school. They did not vote again, but simply crossed out B from all the leaflets. In each sheet there are three candidates left. Therefore, first place was worth 2 points, the second – 1 point, and the third – 0 points. The points were summed up anew.

Could it be that the candidate who previously had the most points, after the self-withdrawal of B received the fewest points?

Gary drew an empty table of $50 \times 50$ and wrote on top of each column and to the left of each row a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down a product of numbers written at the top of its column and to the left of the row $($ the “multiplication table” $)$. What is the largest number of products in this table which could be rational numbers?

A pharmacist has three weights, with which he measured out and gave 100 g of iodine to one buyer, 101 g of honey to another, and 102 g of hydrogen peroxide to the third. He always placed the weights on one side of the scales, and the goods on the other. Could it be that each weight used is lighter than 90 grams?

Replace the letters with numbers $($ all digits must be different $)$ so that the correct equality is obtained: A/ B/ C + D/ E/ F + G/ H/ I = 1.

100 switched on and 100 switched off lights are randomly arranged in two boxes. Each flashlight has a button, the button of which turns off an illuminated flashlight and switches on a turned off flashlight. Your eyes are closed and you can not see if the flashlight is on. But you can move the flashlights from a box to another box and press the buttons on them. Think of a way to ensure that the burning flashlights in the boxes are equally split.

There are 30 students in a class: excellent students, mediocre students and slackers. Excellent students answer all questions correctly, slackers are always wrong, and the mediocre students answer questions alternating one by one correctly and incorrectly. All the students were asked three questions: “Are you an excellent pupil?”, “Are you a mediocre student?”, “Are you a slacker?”. 19 students answered “Yes” to the first question, to the second 12 students answered yes, to the third 9 students answered yes. How many mediocre students study in this class?

Hannah Montana wants to leave the round room which has six doors, five of which are locked. In one attempt she can check any three doors, and if one of them is not locked, then she will go through it. After each attempt her friend Michelle locks the door, which was opened, and unlocks one of the neighbouring doors. Hannah does not know which one exactly. How should she act in order to leave the room?

One hundred cubs found berries in the forest: the youngest managed to grab 1 berry, the next youngest bear – 2 berries, the next – 4 berries, and so on, until the oldest who got $2^{99}$ berries. The fox suggested that they share the berries “in fairness.” She can approach two cubs and distribute their berries evenly between them, and if this leaves an extra berry, then the fox eats it. With such actions, she continues, until all the cubs have an equal number of berries. What is the largest number of berries that the fox can eat?

One hundred cubs found berries in the forest: the youngest managed to grab 1 berry, the next bear in age – 2 berries, the next – 4 berries, and so on, until the oldest bear which found $2^{99}$ berries. The fox suggested that they share the berries in terms of “fairness.” She can approach two cubs and distribute their berries evenly between them, and if this leaves an extra berry, then the fox eats it. With such actions, she continues, until all the cubs have an equal number of berries. What is the smallest number of berries that the fox can leave with the bear cubs?

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. $)$

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?

Catherine laid out 2016 matches on a table and invited Anna and Natasha to play a game which involves taking turns to remove matches from a table: Anna can take 5 matches or 26 matches in her turn, and Natasha can take either 9 or 23. Without waiting for the start of the game, Catherine left, and when she returned, the game was already over. On the table there are two matches, and the one who can not make another turn loses. After a good reflection, Catherine realised which person went first and who won. Figure it out for yourself now.

Hannah recorded the equality $MA \times TE \times MA \times TI \times CA$ = 2016000 and suggested that Charlie replace the same letters with the same numbers, and different letters with different digits, so that the equality becomes true. Does Charlie have the possibility of fulfilling the task?