There are 1,000 inhabitants of Cissylvania. Three of them are vampires, but few know which ones. A visiting writer, Mr. Stocker, asked each resident to name two people he thought were vampires. Each vampire named two other vampires, and the others could name whomever they wanted. Prove that, using the data from the survey (and knowing that there are exactly three vampires in Cissylvania), Mr. Stocker could choose a guide who is not a vampire.

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Seven teams participated in the one-round soccer tournament (each team played exactly one match against each other). At the end of the tournament, the top-ranked teams scored exactly half of the points. Could there have been exactly 6 draws at the end of the tournament? (3 points are given for a win, 1 for a draw and 0 for a loss).

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The grandfather is 56 years old and the grandson is 14.
In how many years will the grandfather be twice as old as the grandson?

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A pack of tea is 50 cents more expensive than a bag of coffee. Oliver bought 7 packs of tea and 6 bags of coffee, spending 68 dollars and 50 cents. How much does a bag of coffee cost?

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Imagine number 45 as a sum of four numbers so that after adding 2 to the first number, subtracting 2 from the second, multiplying by 2 the third, and dividing by 2 the fourth, these numbers will be equal.

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William, Oliver, and James were playing snowballs. The first snowball was thrown by Oliver. Then for every snowball that hit him, William threw six snowballs, James threw five, and Oliver threw four. After a while, the game was over. Find out how many snowballs were hit if 13 snowballs flew past the target. (You do not throw snowballs at yourself.)

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James said on November 28: "Today the difference between the number of full months I have lived and the number of full years I have lived is 144 for the first time. When is James' birthday?

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The figure shows the unfolding of the cube. There are only numbers on it: 1 and 2. Arrange the remaining numbers: 3, 4, 5, 6 — so that the sum of the numbers on any two opposite sides is equal to 7.

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Put 9 points on the plane so that no 4 lie on the same straight line, but out of any 6 there are 3 lying on the same straight line. (In the drawing, draw all the lines on which the three marked points lie.)

📜 Leaflet #geometry

A 2×6 rectangle is marked on the checkered paper.
Is it possible to color the nodes of the cells lying on the border and inside this rectangle (there are 21 of them in total) in two colors so that no four one-color nodes appear at the vertices of the rectangle with sides running along the grid lines?

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There are several natural numbers written in a circle, each of which is not greater than one of its neighbors. Prove that among these numbers there are at least two equal ones.

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There are several numbers written in a circle, each number is equal to the arithmetic mean of two neighboring numbers. Prove that all these numbers are equal.

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Eight mushroom pickers collected 37 mushrooms. It is known that no two of them collected mushrooms equally and each found at least one mushroom. Prove that some two of them collected more than some five.

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There are several cities in the country. A crazy traveler goes from city A to the farthest city B. Then he goes to the farthest from B city C, and so on. Prove that if city C is not the same as city A, then the traveler will never go back to city A.

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There are 2023 asteroids flying in outer space, each of which has an astronomer sitting on it. All distances between asteroids are different. Every astronomer watches the nearest asteroid. Prove that no one is watching one of the asteroids.

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