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?
A box contains 111 red, blue, green, and white marbles. It is known that if we remove 100 marbles from the box, without looking, we will always have removed at least one marble of each colour. What is the minimum number of marbles we need to remove to guarantee that we have removed marbles of 3 different colours?
Author: Shapovalov A.V.
Let \(A\) and \(B\) be two rectangles. From rectangles equal to \(A\), a rectangle similar to \(B\) was created.
Prove that from rectangles equal to \(B\), you can create a rectangle similar to \(A\).
Two play the following game. There is a pile of stones. The first takes either 1 stone or 10 stones with each turn. The second takes either m or n stones with every turn. They take turns, beginning with the first player. He who can not make a move, loses. It is known that for any initial quantity of stones, the first one can always play in such a way as to win (for any strategy of the second player). What values can m and n take?
It is known that \(a > 1\). Is it always true that \(\lfloor \sqrt{\lfloor \sqrt{a}\rfloor }\rfloor = \lfloor \sqrt{4}{a}\rfloor\)?
A robot came up with a cipher for writing words: he replaced some letters of the alphabet with single-digit or two-digit numbers, using only the digits 1, 2 and 3 (different letters it replaces with different numbers). First, he wrote down, using the cipher: \(ROBOT = 3112131233\). Having encrypted the words \(CROCODIL\) and \(BEGEMOT\), he was surprised to note that the numbers were completely identical! Then the Robot ciphered the word \(MATHEMATICS\). Write down the number that he got.
At a contest named “Ah well, monsters!”, 15 dragons stand in a row. Between neighbouring dragons the number of heads differs by 1. If the dragon has more heads than both of his two neighbors, he is considered cunning, if he has less than both of his neighbors – strong, the rest (including those standing at the edges) are considered ordinary. In the row there are exactly four cunning dragons – with 4, 6, 7 and 7 heads and exactly three strong ones – with 3, 3 and 6 heads. The first and last dragons have the same number of heads.
a) Give an example of how this could occur.
b) Prove that the number of heads of the first dragon in all potential examples is the same.
On the left bank of the river, there were 5 physicists and 5 chemists. All of them need to cross to the right bank. There is a two-seater boat. On the right bank at any time there can not be exactly three chemists or exactly three physicists. How do they all cross over by making 9 trips to the right side?