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?
A group of children from two classes came to an after school club: Jack, Ben, Fred, Louis, Claudia, Janine and Charlie. To the question: “How many of your classmates are here?” everyone honestly answered with either “Two” or “Three”. But the boys thought that they were only being asked about the boy classmates, and the girls correctly understood that they were asking about everyone. Is Charlie a boy or a girl?
Hannah recorded the equality \(MA \times TE \times MA \times TI \times CA = 2016000\) and suggested that Charlie replace the same letters with the same numbers, and different letters with different digits, so that the equality becomes true. Does Charlie have the possibility of fulfilling the task?
Catherine laid out 2016 matches on a table and invited Anna and Natasha to play a game which involves taking turns to remove matches from a table: Anna can take 5 matches or 26 matches in her turn, and Natasha can take either 9 or 23. Without waiting for the start of the game, Catherine left, and when she returned, the game was already over. On the table there are two matches, and the one who can not make another turn loses. After a good reflection, Catherine realised which person went first and who won. Figure it out for yourself now.