Problems

Fill the free cells of the “hexagon” (see the figure) with integers from 1 to 19 so that in all vertical and diagonal rows the sum of the numbers, in the same row, is the same.

Is it possible to fill a $5\times 5$ table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?

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.

A teacher filled the squares of a chequered table with $5\times 5$ 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?

• Eight schoolchildren solved $8$ tasks. It turned out that $5$ schoolchildren solved each problem. Prove that there are two schoolchildren, who solved every problem at least once.

• If each problem is solved by $4$ pupils, prove that it is not necessary to have two schoolchildren who would solve each problem.

Is it possible to arrange natural numbers from 1 to ${2002}^2$ in the cells of a $2002\times 2002$ table so that for each cell of this table one could choose a triplet of numbers, from a row or column, where one of the numbers is equal to the product of the other two?

A cinema contains 7 rows each with 10 seats. A group of 50 children went to see the morning screening of a film, and returned for the evening screening. Prove that there will be two children who sat in the same row for both the morning and the evening screening.

A $3\times 3$ square is filled with the numbers $-1,0,+1$. Prove that two of the 8 sums in all directions – each row, column, and diagonal – will be equal.

Some whole numbers are placed into a $10\times 10$ table, so that the difference between any two neighbouring, horizontally or vertically adjacent, squares is no greater than 5. Prove that there will always be two identical numbers in the table.

Is it possible to place the numbers $-1,0,1$ in a $6\times 6$ square such that the sums of each row, column, and diagonal are unique?