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A White Rook pursues a black bishop on a board of 3×1969 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 numbers 1,2,3,,99 are written onto 99 blank cards in order. The cards are then shuffled and then spread in a row face down. The numbers 1,2,3,,99 are once more written onto in the blank side of the cards in order. For each card the numbers written on it are then added together. The 99 resulting summations are then multiplied together. Prove that the result will be an even number.

The sum of 100 natural numbers, each of which is no greater than 100, is equal to 200. Prove that it is possible to pick some of these numbers so that their sum is equal to 100.

The numbers a1,a2,,a1985 are the numbers 1,2,,1985 rearranged in some order. Each number ak is multiplied by its number k, and then the largest number is chosen among the resulting 1985 products. Prove that it is not less than 9932.

The product of 1986 natural numbers has exactly 1985 different prime factors. Prove that either one of these natural numbers, or the product of several of them, is the square of a natural number.

The product of a group of 48 natural numbers has exactly 10 prime factors. Prove that the product of some four of the numbers in the group will always give a square number.

7 different digits are given. Prove that for any natural number n there is a pair of these digits, the sum of which ends in the same digit as the number.

A council of 2,000 deputies decided to approve a state budget containing 200 items of expenditure. Each deputy prepared his draft budget, which indicated for each item the maximum allowable, in his opinion, amount of expenditure, ensuring that the total amount of expenditure did not exceed the set value of S. For each item, the board approves the largest amount of expenditure that is agreed to be allocated by no fewer than k deputies. What is the smallest value of k for which we can ensure that the total amount of approved expenditures does not exceed S?

Prove that there is a number of the form

a) 1989198900 (the number 1989 is repeated several times, and then there are a few zeros), which is divisible by 1988;

b) 19881988, which is divisible by 1989.