Problems

The distance between Athos and Aramis, galloping along one road, is 20 leagues. In an hour Athos covers 4 leagues, and Aramis – 5 leagues.

What will the distance between them be in an hour?

Pinocchio and Pierrot were racing. Pierrot ran the entire race at the same speed, and Pinocchio ran half the way two times faster than Pierrot, and the second half twice as slow as Pierrot. Who won the race?

Find out the principles by which the numbers are depicted in the tables (shown in the figures below) and insert the missing number into the first table, and remove the extra number from the second table.

Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other (it is considered the path of the center of a rook). What is the smallest number of colors to paint the board (each cell is painted with one color), so that two cells, located at a distance of 6, are always painted with different colors?

What is the minimum number of squares that need to be marked on a chessboard, so that:

1) There are no horizontally, vertically, or diagonally adjacent marked squares.

2) Adding any single new marked square breaks rule 1.

Initially, on each cell of a \(1 \times n\) board a checker is placed. The first move allows you to move any checker onto an adjacent cell (one of the two, if the checker is not on the edge), so that a column of two pieces is formed. Then one can move each column in any direction by as many cells as there are checkers in it (within the board); if the column is on a non-empty cell, it is placed on a column standing there and unites with it. Prove that in \(n - 1\) moves you can collect all of the checkers on one square.

The cells of a \(15 \times 15\) square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.

Cut the board shown in the figure into four congruent parts so that each of them contains three shaded cells. Where the shaded cells are placed in each part need not be the same.

In a chess tournament, each participant played two games with each of the other participants: one with white pieces, the other with black. At the end of the tournament, it turned out that all of the participants scored the same number of points (1 point for a victory, \(\frac{1}{2}\) a point for a draw and 0 points for a loss). Prove that there are two participants who have won the same number of games using white pieces.

A teacher filled the squares of a chequered table with \(5\times5\) different integers and gave one copy of it to Janine and one to Zahara. Janine selects the largest number in the table, then she deletes the row and column containing this number, and then she selects the largest number of the remaining integers, then she deletes the row and column containing this number, etc. Zahara performs similar operations, each time choosing the smallest numbers. Can the teacher fill up the table in such a way that the sum of the five numbers chosen by Zahara is greater than the sum of the five numbers chosen by Janine?