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In a row there are 1999 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.

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.

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$)$.

17 squares are marked on an 8×8 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.

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.

12 mayor candidates talked about themselves. After a while, one said: “The number of lies told me before me was one.” Another said: “And now – two”. “And now – three,” said the third, and so on until the 12th, who said: “And now they lied 12 times.” Then the presenter interrupted the discussion. It turned out that at least one candidate correctly calculated how many lies were told before him. So, how many times have the candidates lied?

A cube with side length of 20 is divided into 8000 unit cubes, and on each cube a number is written. It is known that in each column of 20 cubes parallel to the edge of the cube, the sum of the numbers is equal to 1 $($ the columns in all three directions are considered $)$. On some cubes a number 10 is written. Through this cube there are three layers of 1 × 20 × 20 cubes, parallel to the faces of the cube. Find the sum of all the numbers outside of these layers.

Initially, on each cell of a 1 × 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 for the n – 1 move you can collect all of the checkers on one square.

a$)$ Could an additional 6 digits be added to any 6-digit number starting with a 5, so that the 12-digit number obtained is a complete square?

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b$)$ The same question but for a number starting with a 1.

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c$)$ Find for each n the smallest k = k $(n)$ such that to each n-digit number you can assign k more digits so that the resulting $(n + k)$ – digit number is a complete square.

Some real numbers $ a_1, a_2, a_3,…,a _{1996} $ 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.

Two play tic-tac-toe on a 10 × 10 board according to the following rules. First they fill the whole board with noughts and crosses, putting them in turn $($ the first player puts crosses, its partner – noughts $)$. Then two numbers are counted: K is the number of five consecutively standing crosses and H is the number of five consecutively standing zeros. $($ Five, standing horizontally, vertically and parallel to the diagonal are counted, if there are six crosses in a row, this gives two fives, if there are seven, then three, etc.$)$. The number K-H is considered to be the winnings of the first player $($ the losses of the second $)$.

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a$)$ Does the first player have a winning strategy?

b$)$ Does the second player have a winning strategy?

n numbers are given as well as their product, p. The difference between p and each of these numbers is an odd number.

Prove that all n numbers are irrational.

On a plane there is a square, and invisible ink is dotted at a point P. A person with special glasses can see the spot. If we draw a straight line, then the person will answer the question of on which side of the line does P lie $($ if P lies on the line, then he says that P lies on the line $)$.

What is the smallest number of such questions you need to ask to find out if the point P is inside the square?

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.

4 points $a, b, c, d$ lie on the segment $[0, 1]$ of the number line. Prove that there will be a point $x$, lying in the segment $[0, 1]$, that satisfies

$$\frac{1}{\left | x-a\right |}+\frac{1}{\left | x-b\right |}+\frac{1}{\left | x-c\right |}+\frac{1}{\left | x-d\right |} < 40 $$

A 1 × 10 strip is divided into unit squares. The numbers 1, 2, …, 10 are written into squares. First, the number 1 is written in one square, then the number 2 is written into one of the neighboring squares, then the number 3 is written into one of the neighboring squares of those already occupied, and so on $($ the choice of the first square is made arbitrarily and the choice of the neighbor at each step $)$. In how many ways can this be done?

During the ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced with during the previous dance, or more intelligent, but most of the men $($ at least 80% $)$ – with a girl who was at the same time more beautiful and more intelligent. Could this happen? $($ There was an equal number of boys and girls at the ball.$)$

A village infant school has 20 pupils. If we pick any two pupils they will have a shared granddad.

Prove that one of the granddads has no fewer than 14 grandchildren who are pupils at this school.

During a ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced the previous dance with, or more intelligent, and one man danced with a girl who was at the same time both more beautiful and more intelligent. Could this be possible? $($ There was an equal number of both boys and girls $)$.

There is a chocolate bar with five longitudinal and eight transverse grooves, along which it can be broken $($ in total into 9 * 6 = 54 squares $)$. Two players take part, in turns. A player in his turn breaks off the chocolate bar a strip of width 1 and eats it. Another player who plays in his turn does the same with the part that is left, etc. The one who breaks a strip of width 2 into two strips of width 1 eats one of them, and the other is eaten by his partner. Prove that the first player can act in such a way that he will get at least 6 more chocolate squares than the second player.