1. On a chessboard, a knight starts from some square and returns there after making several moves. Show that the knight has made an even number of moves.

2. Can a knight start at a corner square of a chessboard and go to the opposite corner square, visiting each of the remaining squares exactly once on the way?

3. A closed path is made up of 11 line segments. Can one line, not containing a vertex of the path, intersect each of its segments?

4. Three hockey pucks lie on a playing field. A hockey player knocks one of them through the other two. He does this 25 times. Can he return the three pucks to their starting points?

5. Katya and her friends stand in a circle. It turns out that both neighbors of each child are of the same gender. If there are five boys in the circle, how many girls are there?

6. Can we draw a closed path made up of 9 line segments, each of which intersects exactly one of the other segments?

7. Can a 5 X 5 square checkerboard be covered by 1 X 2 dominoes?

8. Given a convex 101-gon which has an axis of symmetry, prove that the axis of symmetry passes through one of its vertices. What can you say about a 10-gon with the same properties?

9. All the dominoes in a set are laid out in a chain (so that the number of spots on the ends of adjacent dominoes match). If one end of the chain is a 5, what is at the other end?

10. In a set of dominoes, all those in which one square has no spots are discarded. Can the remaining dominoes be arranged in a chain?

11. Can a convex 13-gon be divided into parallelograms?

12. Twenty-five checkers are placed on a 25 X 25 checkerboard in such a way that for any i different from j, there is a checker on the square in row i and column j if and only if there is one on the square in row j and column i. Prove that at least one of the cheeckers is on a square with equal row and column numbers.

13. In each square of a 15 X 15 checkerboard, one of the numbers 1, 2, 3, ... , 15 is written. For any i different from j, the number in the square in row i and columnj is equal to that in the square in row j and column i. Moreover, all numbers on each row and on each column are different. Show that all numbers in squares with equal row and column numbers are different.

14. Can one make change of a 25-ruble bill, using in all ten bills each having a value of 1, 3, or 5 rubles?

15. Pete bought a notebook containing 96 pages, and numbered them from 1 through 192. Victor tore out 25 pages of Pete's notebook, and added the 50 numbers he found on the pages. Could Victor have gotten 1990 as the sum?

16. The product of 22 integers is equal to 1. Show that their sum cannot be 0.

17. Can one form a magic square out of the first 36 prime numbers?

18. The numbers 1 through 10 are written in a row. Can the signs + and - be placed between them, so that the value of the resulting expression is 0?

19. A grasshopper jumps along a line. His first jump takes him 1 cm, his second 2 cm, and so on. Each jump can take him to the right or to the left. Show that after 1985 jumps the grasshopper cannot return to the point at which he started.

20. The numbers 1, 2, 3, ... , 1985 are written on a blackboard. We decide to erase from the blackboard any two numbers, and replace them with their difference. After this is done several times, a single number remains on the blackboard. Can this number equal 0?

21. Can a chessboard be covered with 31 dominoes so that only two opposite corner squares remain uncovered?

22. A 17-digit number is chosen, and its digits are reversed, forming a new number. These two numbers are added together. Show that their sum contains at least one even digit.

23. There are 100 soldiers in a detachment, and every evening three of them are on duty. Can it happen that after a certain period of time each soldier has shared duty with every other soldier exactly once?

24. Forty-five points are chosen on line AB, none between A and B. Prove that the sum of the distances from these points to point A is not equal to the sum of the distances from these points to point B.

25. Nine numbers are placed around a circle: four 1's and five 0's. The following operation is performed on the numbers: between each adjacent pair of numbers is placed a 0 if the numbers are different, and a 1 if the numbers are the same. The "old" numbers are then erased. After several of these operations, can all the remaining numbers be equal?

26. Twenty-five boys and twenty-five girls are are seated at a round table. Show that both neighbors of at least one student are boys.

27. A snail crawls along a plane with constant velocity, turning through a right angle every 15 minutes. Show that the snail can return to its starting point only after a whole number of hours.

28. Three grasshoppers play leapfrog along a line. At each turn, one grasshopper leaps over another, but not over two others. Can the grasshoppers return to their initial positions after 1991 leaps?

29. Of 101 coins, 50 are counterfeit, and differ from the genuine coins in weight by 1 gram. Peter has a scale in the form of a balance which shows the difference in weights between the objects placed in each pan. He chooses one coin, and wants to find out in one weighing whether it is counterfeit. Can he do this?

30. Is it possible to arrange the digits 1 through 9 in a sequence so that there are an odd number of digits between 1 and 2, between 2 and 3, ... , and between 8 and 9?