Author: E.V. Bakaev
From the beginning of the academic year, Andrew wrote down his marks for mathematics. When he received another evaluation (2, 3, 4 or 5), he called it unexpected, if before that time this mark was met less often than each of the other possible marks. (For example, if he had received the following marks: 3, 4, 2, 5, 5, 5, 2, 3, 4, 3 from the beginning of the year, the first five and the second four would have been unexpected). For the whole academic year, Andrew received 40 marks - 10 fives, fours, threes and twos (it is not known in which order). Is it possible to say exactly how many marks were unexpected?
To a certain number, we add the sum of its digits and the answer we get is 2014. Give an example of such a number.
Author: E.V. Bakaev
After a hockey match Anthony said that he scored 3 goals, and Ilya only one. Ilya said that he scored 4 goals, and Serge scored 5 goals. Serge said that he scored 6 goals, and Anthony only two. Could it be that the three of them scored 10 goals, if it is known that each of them once told the truth, and once lied?
Find all of the solutions of the puzzle: \(ARKA + RKA + KA + A = 2014\). (Different letters correspond to different numbers, and the same letters correspond to the same numbers.)
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?
Anna is waiting for the bus. Which event is most likely?
\(A =\{\)Anna waits for the bus for at least a minute\(\}\),
\(B = \{\)Anna waits for the bus for at least two minutes\(\}\),
\(C = \{\)Anna waits for the bus for at least five minutes\(\}\).
Peter and 9 other people play such a game: everyone rolls a dice. The player receives a prize if he or she rolled a number that no one else was able to roll.
a) What is the probability that Peter will receive a prize?
b) What is the probability that at least someone will receive a prize?
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)?
Hannah Montana wants to leave the round room which has six doors, five of which are locked. In one attempt she can check any three doors, and if one of them is not locked, then she will go through it. After each attempt her friend Michelle locks the door, which was opened, and unlocks one of the neighbouring doors. Hannah does not know which one exactly. How should she act in order to leave the room?