Know-it-all came to visit the twin brothers Screw and Nut, knowing that one of them never speaks the truth, and asked one of them: “Are you Screw?”. “Yes,” he replied. When Know-it-all asked the second brother the same question, he received an equally clear answer and immediately determined who was who.
Who was called Screw?
Solve the puzzle: \(BAO \times BA \times B = 2002\).
The numbers from 1 to 9999 are written out in a row. How can I remove 100 digits from this row so that the remaining number is a) maximal b) minimal?
The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?
Four friends came to an ice-rink, each with her brother. They broke up into pairs and started skating. It turned out that in each pair the “gentleman” was taller than the “lady” and no one is skating with his sister. The tallest boy in the group was Sam Smith, Peter Potter, then Luisa Potter, Joe Simpson, Laura Simpson, Dan Caldwell, Jane Caldwell and Hannah Smith. Determine who skated with whom.
In the race of six athletes, Andrew lagged behind Brian and two more athletes. Victor finished after Dennis, but before George. Dennis beat Brian, but still came after Eustace. What place did each athlete take?
We are given 101 rectangles with integer-length sides that do not exceed 100.
Prove that amongst them there will be three rectangles \(A, B, C\), which will fit completely inside one another so that \(A \subset B \subset C\).