Let’s prove that \(1=2\). Take a number \(a\) and suppose \(b=a\). After multiplying both sides we have \(a^2=ab\). Subtract \(b^2\) from both sides to get \(a^2-b^2=ab-b^2\). The left hand side is a difference of two squares so \((a-b)(a+b)=b(a-b)\). We can cancel out \(a-b\) and obtain that \(a+b=b\). But remember from the start that \(a=b\), so substituting \(a\) for \(b\) we see that \(2b=b\), dividing by \(b\) we see that \(2=1\).
John drunk a \(\frac16\) of a full cup of black coffee and then filled the cup back up with milk. Then he drunk a third of what he had in the cup. Then, he refilled it back to full with milk again, and after that, drunk a half of the cup. Finally, he once again refilled the cup with milk and drunk everything he had. What did he drink more of - coffee or milk?
A round necklace contains \(45\) beads of two different colours: red and blue. Show that it is possible to find two beads of the same colour next to each other.
A broken calculator can only do several operations: multiply by \(2\), divide by \(2\), multiply by \(3\), divide by \(3\), multiply by \(5\), and divide by \(5\). Using this calculator any number of times, could you start with the number \(12\) and end up with \(49\)?
The numbers \(1\) through \(12\) are written on a board. You can erase any two of these numbers (call them \(a\) and \(b\)) and replace them with the number \(a+b-1\). Notice that in doing so, you remove one number from the total, so after \(11\) such operations, there will be just one number left. What could this number be?
There are real numbers written on each field of a \(m \times n\) chessboard. Some of them are negative, some are positive. In one move we can multiply all the numbers in one column or row by \(-1\). Is that always possible to obtain a chessboard where sums of numbers in each row and column are non-negative?
Tom found a large, old clock face and put \(12\) sweets on the number \(12\). Then he started to play a game: in each move he moves one sweet to the next number clockwise, and some other to the next number anticlockwise. Is it possible that after finite number of steps there is exactly \(1\) of the sweets on each number?
Anna has \(20\) novels and \(25\) comic books on her shelf. She doesn’t really keep her room very tidy and so she also has a lot of novels and comic books in various places around her room. Each time she reaches for the shelf, she takes two books and puts one back. If she takes two novels or two comic books, she puts a novel back on the shelf. If she takes a novel and a comic book, she places another comic book on the shelf. That way Anna’s shelf systematically empties, since after every operation there is one book less. Show that eventually there will be a lone comic book standing on her shelf and all her other books scattered across her room.
A knight in chess moves in an "\(L\)" pattern - two squares in one direction and one square in a perpendicular direction. Starting in the bottom right corner of a regular \(8 \times 8\) chessboard, is it possible for a knight to visit every square on the chessboard exactly once and end up in the top left corner?
Nine lightbulbs are arranged in a \(3 \times 3\) square. Some of them are on, some are off. You are allowed to change the state of all the bulbs in a column or in a row. That means all the bulbs in that row or column that were initially off now light up, and the ones that were initially on, now go dark. Is it possible to go from the arrangement in the left to the one on the right by repeating this operation?