A group of psychologists developed a test, after which each person gets a mark, the number \(Q\), which is the index of his or her mental abilities (the greater \(Q\), the greater the ability). For the country’s rating, the arithmetic mean of the \(Q\) values of all of the inhabitants of this country is taken.
a) A group of citizens of country \(A\) emigrated to country \(B\). Show that both countries could grow in rating.
b) After that, a group of citizens from country \(B\) (including former ex-migrants from \(A\)) emigrated to country \(A\). Is it possible that the ratings of both countries have grown again?
c) A group of citizens from country \(A\) emigrated to country \(B\), and group of citizens from country \(B\) emigrated to country \(C\). As a result, each country’s ratings was higher than the original ones. After that, the direction of migration flows changed to the opposite direction – part of the residents of \(C\) moved to \(B\), and part of the residents of \(B\) migrated to \(A\). It turned out that as a result, the ratings of all three countries increased again (compared to those that were after the first move, but before the second). (This is, in any case, what the news agencies of these countries say). Can this be so (if so, how, if not, why)?
(It is assumed that during the considered time, the number of citizens \(Q\) did not change, no one died and no one was born).
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?
Each of the 1994 deputies in parliament slapped exactly one of his colleagues. Prove that it is possible to draw up a parliamentary commission of 665 people whose members did not clarify the relationship between themselves in the manner indicated above.
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.
We are given a \(100\times 100\) square grid and \(N\) counters. All of the possible arrangements of the counters on the grid which follow the following rule are considered: no two counters lie in adjacent squares.
What is the largest value of \(N\) for which, in every single possible arrangement of counters following this rule, it is possible to find at least one counter such that moving it to an adjacent square does not break the rule. Squares are considered adjacent if they share a side.
On a particular day it turned out that every person living in a particular city made no more than one phone call. Prove that it is possible to divide the population of this city into no more than three groups, so that within each group no person spoke to any other by telephone.
We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.
Is it possible to arrange natural numbers from 1 to \(2002^2\) in the cells of a \(2002\times2002\) 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?
We are given a table of size \(n \times n\). \(n-1\) of the cells in the table contain the number \(1\). The remainder contain the number \(0\). We are allowed to carry out the following operation on the table:
1. Pick a cell.
2. Subtract 1 from the number in that cell.
3. Add 1 to every other cell in the same row or column as the chosen cell.
Is it possible, using only this operation, to create a table in which all the cells contain the same number?
Prove that in any set of 117 unique three-digit numbers it is possible to pick 4 non-overlapping subsets, so that the sum of the numbers in each subset is the same.