Use the extreme principle to solve the following problem: There are 2000 points on a circle, and each point is given a number which is equal to the average of the numbers of its two nearest neighbors. Show that all the numbers must be equal.

Place the integers 1, 2, 3, \ldots , n^2 without duplication on a n \times n chessboard, with one integer per square. Show that there exists two adjacent entries whose difference is at least n+1. (Adjacent means horizontally, vertically or diagonally adjacent.)

Five Houses On a certain street there are five houses in a row. Each occupant of each house made a number of statements. They follow:

1. Mr Archer says that Mr Beadle lives next door to him, and Mrs Dickens does not live next door to Mr Beadle. The two women live in adjacent houses, he adds.
2. Mr Beadle says that Mrs Carver lives next door to Mr Ezekiel, and the rightmost house is occupied by Mrs Dickens.
3. Mrs Caver says that Mrs Dickens tell the truth, she (Mrs Carver) lives in the middle house, and Mr Archer lives in the first house.
4. Mrs Dickens says that she lives to the left of Mr. Ezekiel and Mr Beadle does not live next door to Mr. Ezekiel.
5. Mr. Ezekiel say she lives next door to Mrs Caver. He says he lives to the left of Mr Beadle, and Mrs Dickens lives at one end of the street.

Unfortunately it is unknown who tells the truth of lies; however, all the residents either tell the truth consistently of lie consistently. Who lives in what house? Find all possible answers when &dlquo;to the left of&drquo; means &dlquo;somewhere to the left of.&drquo;