In a burrow there is a family of 24 mice. Every night exactly four of them are sent to the warehouse for cheese.
Could it occur that at some point in time each mouse went to the warehouse with every other mouse exactly one time?
Let \(x\) be a 2 digit number. Let \(A\), \(B\) be the first (tens) and second (units) digits of \(x\), respectively. Suppose \(A\) is twice as large as \(B\). If we add the square of \(A\) to \(x\) then we get the square of a certain whole number. Find the value of \(x\).
In a physics club, the teacher created the following experiment. He spread out 16 weights of weight 1, 2, 3, ..., 16 grams onto weighing scales, so that one of the bowls outweighed the other. Fifteen students in turn left the classroom and took with them one weight each, and after each student’s departure, the scales changed their position and outweighed the opposite bowl of the scales. What weight could remain on the scales?
There is a \(5\times 9\) rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?
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 \(a\).
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 \(n \geq k\).
The bank of the Nile was approached by a group of six people: three Bedouins, each with his wife. At the shore is a boat with oars, which can withstand only two people at a time. A Bedouin can not allow his wife to be without him whilst in the company of another man. Can the whole group cross to the other side?
Is it possible for the mean of some 35 whole numbers to equal \(6.35\)?