The grasshopper jumps in a straight line 1 meter to the left or to the right each time. After some time he is at the starting point. Prove that he has made an even number of jumps.

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All dice with "blanks" have been discarded from the domino set. Can the remaining dice be placed in a row according to the rules?

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There are boys and girls sitting at the round table. Prove that the number of pairs of boy-girl and girl-boy neighbors is even.

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A snail crawls along a plane at a constant speed, turning 90° every 30 minutes. Prove that it can return to its starting point only:
1) after an integer number of hours;
2) after an even number of hours.

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There are 8 rooks on the chessboard, of which no two beat each other. Prove that the number of rooks standing on the black fields is even.

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To a 17-digit number add a number written with the same digits, but in reverse order. Prove that at least one digit of the resulting sum is even.

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There are 5 kinds of cups, 4 kinds of saucers, and 2 kinds of spoons in the "Everything for Tea" store. How many ways can you buy in this store:
1) a set of a cup, saucer, and spoon?
2) a set consisting of two different items?

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Wikipedia

Let's call a natural number "cute" if only even digits occur in its record. How many four-digit "cute" numbers are there?

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A soccer team needs to choose a captain and a deputy captain. How many ways can this be done? (There are 11 players on the soccer team.)

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There are 25 students in the class. How many ways can you choose two people to be on duty?

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The alphabet of the Mumbo Yumbo tribe consists of three letters: A, B and C. A word is any sequence of no more than 4 letters. How many words are there in the Mumbo Yumbo language?

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The menu in the school cafeteria does not change and consists of 10 dishes. For a change Oliver wants to order every day a set of dishes that he has never ordered before (the number of dishes does not matter, he can order all 10 dishes, or he can order only one, or none at all). How many days can he eat like that?

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Write out all sets of three digits, each of which is equal to 1, 2 or 3, if the order of the digits is unimportant (i.e. sets 112 and 121 are considered the same).

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Wikipedia

The box contains blue, red, and green pencils. There are 20 in all. There are 6 times as many blue pencils as green pencils, and fewer red pencils than blue. How many red pencils are in the box?

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There are dominoes in the box. The figure shows only how the halves of the dominoes are arranged, but the borders are not shown. Determine how they pass.

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List all the four natural numbers giving a total of 15.

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A flock of one-headed centipedes and three-headed dragons was flying. Together they had 648 legs and 39 heads. How many legs does a dragon have?

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Cut the drawn figure into two identical (matching when superimposed) parts.

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There are 17 parallels and 24 meridians on the globe.
How many parts is the surface of the globe divided into?

ℹ️ Meridian is an arc connecting the North Pole with the South Pole.
Parallel is a circle parallel to the equator (the equator is also a parallel).

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