Problems

Some points with integer co-ordinates are marked on a Cartesian plane. It is known that no four points lie on the same circle. Prove that there will be a circle of radius 1995 in the plane, which does not contain a single marked point.

Some real numbers $a_1,a_2,a_3,\dots ,a_{2022}$ are written in a row. Prove that it is possible to pick one or several adjacent numbers, so that their sum is less than 0.001 away from a whole number.

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.

In a regular shape with 25 vertices, all the diagonals are drawn.

Prove that there are no nine diagonals passing through one interior point of the shape.

17 squares are marked on an $8\times 8$ chessboard. In chess a knight can move horizontally or vertically, one space then two or two spaces then one – eg: two down and one across, or one down and two across. Prove that it is always possible to pick two of these squares so that a knight would need no less than three moves to get from one to the other.

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 square is cut by 18 straight lines, 9 of which are parallel to one side of the square and the other 9 parallel to the other – perpendicular to the first 9 – dividing the square into 100 rectangles. It turns out that exactly 9 of these rectangles are squares. Prove that among these 9 squares there will be two that are identical.

In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.

2022 dollars were placed into some wallets and the wallets were placed in some pockets. It is known that there are more wallets in total than there are dollars in any pocket. Is it true that there are more pockets than there are dollars in one of the wallets? You are not allowed to place wallets one inside the other.

In a basket, there are 30 mushrooms. Among any 12 of them there is at least one brown one, and among any 20 mushrooms, there is at least one chanterelle. How many brown mushrooms and how many chanterelles are there in the basket?