Author: A.V. Khachaturyan
The mum baked some pies – three with peach, three with kiwi and one with blackberries – and laid them on the dish in a circle (see the picture). Then she put the dish in a microwave to warm it up. All of the pies look the same. Maria knows how they lie on the dish but does not know how the dish turned in the microwave. She wants to eat a pie with blackberries, but she doesn’t want any of the others because she doesn’t like their taste. How can Maria surely achieve this by biting as few tasteless pies as possible?
A disk contains 2013 files of 1 MB, 2 MB, 3 MB, …, 2012 MB, 2013 MB. Can I distribute them in three folders so that each folder has the same number of files and all three folders have the same size $($in MB$)$?
Author: D.V. Baranov
Vlad and Peter are playing the following game. On the board two numbers written are: 1/2009 and 1/2008. At each turn, Vlad calls any number x, and Peter increases one of the numbers on the board (whichever he wants) by x. Vlad wins if at some point one of the numbers on the board becomes equal to 1. Will Vlad win, no matter how Peter acts?
The sheikh spread out his treasures in nine sacks: 1 kg in the first bag, 2 kg in the second bag, 3 kg in the third bag, and so on, and 9 kg in the ninth bag. The insidious official stole a part of the treasure from one bag. How can the sheikh work out from which bag the official stole the treasure from using two weighings?
All of the inhabitants of the island are either knights and speak only the truth, or liars and always lie. A traveller met five islanders. To his question: “How many knights are among you?” the first replied: “None!”, and the other two answered: “One.” What did the others say?
Of five coins, two are fake. One of the counterfeit coins is lighter than the real one, and the other is heavier than the real one by as much as the lighter one is lighter than the real coin.
Explain how in the three weighings, you can find both fake coins using scales without weights.
The island has 100 knights and 100 liars. Each of them has at least one friend. Once exactly 100 people said: “All my friends are knights,” and exactly 100 people said: “All my friends are liars.” What is the smallest possible number of pairs of friends, one of whom is a knight and the other a liar?
There are 13 pupils in the school of witchcraft and wizardry. Before the Clairvoyance exam, the teacher put them at a round table and asked to guess who would receive the clairvoyant’s diploma. The students said nothing about themselves and two of their neighbours, but they wrote the following about all of the others: “None of these ten will get the diploma!” Of course, all of those who passed the exam guessed correctly, and all of the other students were mistaken. How many wizards received a diploma?
Author: I.V. Izmestyev
Postman Pat did not want to give away the parcel. So, Matt suggested that he play the following game: every move, Pat writes in a line from left to right the letters M and P, randomly alternating them, until he has a line made up of 11 letters. Matt, after each of Pat’s moves, if he wants, swaps any two letters. If in the end it turns out that the recorded word is a palindrome $($that is, it is the same if read from left to right and right to left$)$, then Pat gives Matt the parcel. Can Matt play in such a way as to get the parcel?
Author: D.E. Shnol
On the island of Truthland, all of the inhabitants may be mistaken, but the younger ones never contradict the elders, and when the older ones contradict the younger ones, they $($the elders$)$ are not mistaken. Between the residents A, B and C there was such a conversation:
A: B is the tallest.
B: A is the tallest.
C: I’m taller than B.
Does it follow from this conversation that the younger the person, the taller he or she is $($for the three people having this conversation$)$?
There are 12 people in a room. Some of them are honest (i.e. always tell the truth) and the rest always lie. “There are no honest people here,” said the first person. “There is not more than one honest person here,” said the second person. The third person said that there are no more than 2 honest people, the fourth person said there are no more than 3 honest people, and so on until the twelfth person, who said there are no more than 11 honest people. How many honest people are in the room?
There are 4 coins. Of the four coins, one is fake $($it differs in weight from the real ones, but it is not known if it is heavier or lighter$)$. Find the fake coin using two weighings on scales without weights.
Peter thought of a number between 1 to 200. What is the fewest number of questions for which you can guess the number if Peter answers
a) “yes ” or “no”;
b) “yes”, “no” or “I do not know”
for every question?
a) One person had a basement illuminated by three electric bulbs. Switches of these bulbs are located outside the basement, so that having switched on any of the switches, the owner has to go down to the basement to see which lamp switches on. One day he came up with a way to determine for each switch which bulb it switched on, descending into the basement exactly once. What is the method?
b) If he goes down to the basement exactly twice, how many bulbs can he identify the switches for?
How many integers are there from 1 to 1,000,000, which are neither full squares, nor full cubes, nor numbers to the fourth power?
Each side in the triangle ABC is divided into 8 equal segments. How many different triangles exist with the vertices at the points of division $($the points A, B, C cannot be the vertices of triangles$)$ in which neither side is parallel to either side of the triangle ABC?
A rectangular table is given, in each cell of which a real number is written, and in each row of the table the numbers are arranged in ascending order. Prove that if you arrange the numbers in each column of the table in ascending order, then in the rows of the resulting table, the numbers will still be in ascending order.
A message is encrypted using numbers where each number corresponds to a different letter of the alphabet. Decipher the following encoded text:
Each of the three axes has one rotating pin and a fixed arrow. The gears are connected in series. On the first gear there are 33 teeth, on the second – 10, on the third – 7. On each tooth of the first gear one symbol or letter of the following string of letters and symbols is written in the clockwise direction in the following order:
A B V C D E F G H I J K L M N O P Q R S T U W X Y Z ! ? > < $ £ €
On the teeth of the second and third gears in increasing order the numbers 0 to 9 and 0 to 6 are written respectively in a clockwise direction. When the arrow of the first axis points to a letter, the arrows of the other two axes point to numbers.
The letters and symbols of the message are encrypted in sequence. Encryption is performed by rotating the first gear anti-clockwise until the first possible letter or symbol that can be encrypted is landed on by the arrow. At this point, the numbers indicated by the second and third arrows are consistently written out. At the beginning of the encryption, the 1st wheel points to the letter A, and the arrows of the 2nd and 3rd wheels to the number 0.
Encrypt the Slavic name OLIMPIADA.
The key of the cipher, called the “lattice”, is a rectangular stencil of size 6 by 10 cells. In the stencil, 15 cells are cut out so that when applied to a rectangular sheet of paper of size 6 by 10, its cut-outs completely cover the entire area of the sheet in four possible ways. The letters of the string $($without spaces$)$ are successively entered into the cut-outs of the stencil $($in rows, in each line from left to right$)$ at each of its four possible positions. Find the original string of letters if, after encryption, the following text appeared in the sheet of paper