Problems

In a board, 20 pins are placed (see the picture). The distance between any adjacent pins is 1 inch. Pull a string of length 19 inches from the first pin to the second one, so that it goes through all the pins.

Henry did not manage to get into the elevator on the first floor of the building and decided to go up the stairs. It takes 2 minutes to rise to the third floor. How long does it take to rise to the ninth floor?

There are \(12\) aliens in the High Council of the planet of liars and truth tellers. “There is no-one honest here,” said the first member of the council. “There is at most one honest person here,” said the second person. The third person said that there are at most \(2\) honest members, the fourth person said there are at most \(3\) honest aliens, and so on until the twelfth person, who said there are at most \(11\) honest aliens. How many honest members are in the High Council?

In a chess tournament, each participant played two games with each of the other participants: one with white pieces, the other with black. At the end of the tournament, it turned out that all of the participants scored the same number of points (1 point for a victory, \(\frac{1}{2}\) a point for a draw and 0 points for a loss). Prove that there are two participants who have won the same number of games using white pieces.

The grasshopper jumps on the interval \([0,1]\). On one jump, he can get from the point \(x\) either to the point \(x/3^{1/2}\), or to the point \(x/3^{1/2} + (1- (1/3^{1/2}))\). On the interval \([0,1]\) the point \(a\) is chosen.

Prove that starting from any point, the grasshopper can be, after a few jumps, at a distance less than \(1/100\) from point \(a\).

Two different numbers \(x\) and \(y\) (not necessarily integers) are such that \(x^2-2000x=y^2-2000y\). Find the sum of \(x\) and \(y\).

In a mathematical olympiad, \(m>1\) candidates solved \(n>1\) problems. Each candidate solved a different number of problems to all the others. Each problem was solved by a different number of candidates to all the others. Prove that one of the candidates solved exactly one problem.

A teacher filled the squares of a chequered table with \(5\times5\) different integers and gave one copy of it to Janine and one to Zahara. Janine selects the largest number in the table, then she deletes the row and column containing this number, and then she selects the largest number of the remaining integers, then she deletes the row and column containing this number, etc. Zahara performs similar operations, each time choosing the smallest numbers. Can the teacher fill up the table in such a way that the sum of the five numbers chosen by Zahara is greater than the sum of the five numbers chosen by Janine?

All of the sweets of different sorts in stock are arranged in \(n\) boxes, for which prices are set at \(1, 2, \dots , n\), respectively. It is required to buy such \(k\) of these boxes of the least total value, which contain at least \(k/n\) of the mass of all of the sweets. It is known that the mass of sweets in each box does not exceed the mass of sweets in any more expensive box.

a) What boxes should I buy when \(n = 10\) and \(k = 3\)?

b) The same question for arbitrary natural numbers \(n \geq k\).

Is it possible for the mean of some 35 whole numbers to equal \(6.35\)?