In a row there are 20 different natural numbers. The product of every two of them standing next to one another is the square of a natural number. The first number is 42. Prove that at least one of the numbers is greater than 16,000.
There are scales and 100 coins, among which several (more than 0 but less than 99) are fake. All of the counterfeit coins weigh the same and all of the real ones also weigh the same, while the counterfeit coin is lighter than the real one. You can do weighings on the scales by paying with one of the coins (whether real or fake) before weighing. Prove that it is possible with a guarantee to find a real coin.
Author: I.S. Rubanov
On the table, there are 7 cards with numbers from 0 to 6. Two take turns in taking one card. The winner is the one is the first person who can, from his cards, make up a natural number that is divisible by 17. Who will win in a regular game the person who goes first or second?
At a round table, 2015 people are sitting down, each of them is either a knight or a liar. Knights always tell the truth, liars always lie. They were given one card each, and on each card a number is written; all the numbers on the cards are different. Looking at the cards of their neighbours, each of those sitting at the table said: “My number is greater than that of each of my two neighbors.” After that, \(k\) of the sitting people said: “My number is less than that of each of my two neighbors.” At what maximum \(k\) could this occur?
a) There are three identical large vessels. In one there are 3 litres of syrup, in the other – 20 litres of water, and the third is empty. You can pour all the liquid from one vessel into another or into a sink. You can choose two vessels and pour into one of them liquid from the third, until the liquid levels in the selected vessels are equal. How can you get 10 litres of diluted 30% syrup?
b) The same, but there is \(N\) l of water. At what integer values of \(N\) can you get 10 liters of diluted 30% syrup?
It is known that \(AA + A = XYZ\). What is the last digit of the product: \(B \times C \times D \times D \times C \times E \times F \times G\) (where different letters denote different digits, identical letters denote identical digits)?
Monica is in a broken space buggy at a distance of 18 km from the Lunar base, in which Rachel sits. There is a stable radio communication system between them. The air reserve in the space buggy is enough for 3 hours, in addition, Monica has an air cylinder for the spacesuit, with an air reserve of 1 hour. Rachel has a lot of cylinders with an air supply of 2 hours each. Rachel can not carry more than two cylinders at the same time (one of them she uses herself). The speed of movement on the Moon in the suit is 6 km/h. Could Rachel save Monica and not die herself?
There are 40 identical cords. If you set any cord on fire on one side, it burns, and if you set it alight on the other side, it will not burn. Ahmed arranges the cords in the form of a square (see the figure below, each cord makes up a side of a cell). Then, Helen arranges 12 fuses. Will Ahmed be able to lay out the cords in such a way that Helen will not be able to burn all of them?
301 schoolchildren came to the school’s New Year’s party in the city of Moscow. Some of them always tell the truth, and the rest always lie. Each of some 200 students said: “If I leave the hall, then among the remaining students, the majority will be liars.” Each of the other schoolchildren said: “If I leave the room, then among the remaining students, there will be twice as many liars as those who speak the truth.” How many liars were at the party?
The pupils of class 5A had a total of 2015 pencils. One of them lost a box with five pencils, and instead bought a box with 50 pencils. How many pencils do the pupils of class 5A now have?