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?
A castle is surrounded by a circular wall with nine towers, at which there are knights on duty. At the end of each hour, they all move to the neighbouring towers, each knight moving either clockwise or counter-clockwise. During the night, each knight stands for some time at each tower. It is known that there was an hour when at least two knights were on duty at each tower, and there was an hour when there was precisely one knight on duty on each of exactly five towers. Prove that there was an hour when there were no knights on duty on one of the towers.
In the entry \({*} + {*} + {*} + {*} + {*} + {*} + {*} + {*} = {*}{*}\) replace the asterisks with different digits so that the equality is correct.
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)?
Hannah placed 101 counters in a row which had values of 1, 2 and 3 points. It turned out that there was at least one counter between every two one point counters, at least two counters lie between every two two point counters, and at least three counters lie between every two three point counters. How many three point counters could Hannah have?
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?