A family went to the bridge at night. The dad can cross over it in 1 minute, the mom can cross it in 2, the child takes 5 minutes, and grandmother in 10 minutes. They have one flashlight. The bridge can only withstands two people at a time. How can they all cross the bridge in 17 minutes? (If two people pass, then they go at the lower of their speeds.) You can not move along a bridge without a flashlight. You can not shine it from a distance.
On the island of Contrast, both knights and liars live. Knights always tell the truth, liars always lie. Some residents said that the island has an even number of knights, and the rest said that the island has an odd number of liars. Can the number of inhabitants of the island be odd?
In Mongolia there are in circulation coins of 3 and 5 tugriks. An entrance ticket to the central park costs 4 tugriks. One day before the opening of the park, a line of 200 visitors queued up in front of the ticket booth. Each of them, as well as the cashier, has exactly 22 tugriks. Prove that all of the visitors will be able to buy a ticket in the order of the queue.
There is a rectangular table. Two players start in turn to place on it one pound coin each, so that these coins do not overlap one another. The player who cannot make a move loses. Who will win with the correct strategy?
a) Two players play in the following game: on the table there are 7 two pound coins and 7 one pound coins. In a turn it is allowed to take coins worth no more than three pounds. The one who takes the last coin wins. Who will win with the correct strategy?
b) The same question, if there are 12 one pound and 12 two pound coins.
On Easter Island, people ask each other questions, to which only “yes” or “no” can be answered. In this case, each of them belongs exactly to one of the tribes either A or B. People from tribe A ask only those questions to which the correct answer is “yes”, and from tribe B – those questions to which the correct answer is “no.” In one house lived a couple Ethan and Violet Russell. When Inspector Krugg approached the house, the owner met him on the doorstep with the words: “Tell me, do Violet and I belong to tribe B?”. The inspector thought and gave the right answer. What was the right answer?
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?
Find all functions \(f (x)\) such that \(f (2x + 1) = 4x^2 + 14x + 7\).
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 \(n \geq k\).