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.
A traveller met five inhabitants of the planet of liars and truth tellers. To his question: “How many truth tellers are there among you?” the first replied: “None!", and another two answered: “One.” What did the final two say?
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?
The height of the room is 3 meters. When it was being renovated, it turned out that more paint was needed on each wall than on the floor. Can the area of the floor of this room be more than 10 square meters?
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?
Find the largest value of the expression \(a + b + c + d - ab - bc - cd - da\), if each of the numbers \(a\), \(b\), \(c\) and \(d\) belongs to the interval \([0, 1]\).
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: 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?
Author: A.V. Khachaturyan
Replace the letters of the word \(MATEMATIKA\) with numbers and signs of addition and subtraction so that a numeric expression equal to 2014 is obtained.
(The same letters denote the same numbers or signs, different letters denote different numbers or signs. Note that it is enough to give an example.)
Author: N.K. Agakhanov
On the board, the equation \(xp^3 + * x^2 + * x + * = 0\) is written. Peter and Vlad take turns to replace the asterisks with rational numbers: first, Peter replaces any of the asterisks, then Vlad – any of the two remaining ones, and then Peter replaces the remaining asterisk. Is it true that for any of Vlad’s actions, Peter can get an equation in which the difference of some two roots is equal to 2014?