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Among 4 people there are no three with the same name, the same middle name and the same surname, but any two people have either the same first name, middle name or surname. Can this be so?

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) We are given two cogs, each with 14 teeth. They are placed on top of one another, so that their teeth are in line with one another and their projection looks like a single cog. After this 4 teeth are removed from each cog, the same 4 teeth on each one. Is it always then possible to rotate one of the cogs with respect to the other so that the projection of the two partially toothless cogs appears as a single complete cog? The cogs can be rotated in the same plane, but cannot be flipped over.

b) The same question, but this time two cogs of 13 teeth each from which 4 are again removed?

The function F is given on the whole real axis, and for each x the equality holds: F $(x + 1)$ F $(x)$ + F $(x + 1)$ + 1 = 0.

Prove that the function F can not be continuous.

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 centre of a rook$)$. What is the smallest number of colours to paint the board $($each cell is painted with one colour$)$, so that two cells, located at a distance of 6, are always painted with different colours?

The smell of a flowering lavender plant diffuses through a radius of 20m around it. How many lavender plants must be planted along a straight 400m path so that the smell of the lavender reaches every point on the path.

The peasant must transport across a river a wolf, a goat and a cabbage. The boat accommodates one person, and with him/her either a wolf, a goat, or a cabbage. If you leave the goat and the wolf unattended, the wolf will eat the goat. If you leave cabbage and goat without supervision, the goat will eat the cabbage. How can the peasant transport his cargo across the river?

Does there exist a number h such that for any natural number n the number [$h \times 1969^n$] is not divisible by [$h \times 1969^{n-1}$]?

What is the largest amount of numbers that can be selected from the set 1, 2, …, 1963 so that the sum of any two numbers is not divisible by their difference?

A table of $4\times4$ cells is given, in some cells of which a star is placed. Show that you can arrange seven stars so that when you remove any two rows and any two columns of this table, there will always be at least one star in the remaining cells. Prove that if there are fewer than seven stars, you can always remove two rows and two columns so that all the remaining cells are empty.

The equations $ax^2$ + bx + c = 0 $(1)$ and – $ax^2$ + bx + c $(2)$ are given. Prove that if $x_1$ and $x_2$ are, respectively, any roots of the equations $(1)$ and $(2)$, then there is a root $x_3$ of the equation ½ $ax^2$ + bx + c such that either $x_1$ $≤$ $x_3$ $≤$ $x_2$ or $x_1$ $≥$ $x_3$ $≥$ $x_2$.

The triangle $C_1C_2O$ is given. Within it the bisector $C_2C_3$ is drawn, then in the triangle $C_2C_3O$ – bisector $C_3C_4$ and so on. Prove that the sequence of angles $γ_n$ = $C_{n + 1}C_nO$ tends to a limit, and find this limit if $C_1OC_2$ = α.

Author: V.A. Popov

On the interval [0; 1] a function f is given. This function is non-negative at all points, f $(1)$ = 1 and, finally, for any two non-negative numbers $x_1$ and $x_2$ whose sum does not exceed 1, the quantity $f (x_1 + x_2)$ does not exceed the sum of $f (x_1)$ and $f (x_2 )$.

a) Prove that for any number x on the interval [0; 1], the inequality $f (x_2) ≤ 2x$ holds.

b) Prove that for any number x on the interval [0; 1], the $f (x_2) ≤ 1.9x$ must be true?

A grasshopper can make jumps of 8, 9 and 10 cells in any direction on a strip of n cells. We will call the natural number n jumpable if the grasshopper can, starting from some cell, bypass the entire strip, having visited each cell exactly once. Find at least one n $>$ 50 that is not jumpable.

We took several positive numbers and constructed the following sequence: $a_1$ is the sum of the initial numbers, $a_2$ is the sum of the squares of the original numbers, $a_3$ is the sum of the cubes of the original numbers, and so on.

a$)$ Could it happen that up to $a_5$ the sequence decreases $(a_1> a_2> a_3> a_4> a_5)$, and starting with $a_5$ – it increases $(a_5 < a_6 < a_7 <...)$?

b$)$ Could it be the other way around: before a_5 the sequence increases, and starting with a_5 – decreases?

Three cyclists travel in one direction along a circular track that is 300 meters long. Each of them moves with a constant speed, with all of their speeds being different. A photographer will be able to make a successful photograph of the cyclists, if all of them are on some part of the track which has a length of d meters. What is the smallest value of d for which the photographer will be able to make a successful photograph sooner or later?

In each cell of a board of size $5\times5$ a cross or a nought is placed, and no three crosses are positioned in a row, either horizontally, vertically or diagonally. What is the largest number of crosses on the board?

In a tournament, 100 wrestlers are taking part, all of whom have different strengths. In any fight between two wrestlers, the one who is stronger always wins. In the first round the wrestlers broke into random pairs and fought each other. For the second round, the wrestlers once again broke into random pairs of rivals $($it could be that some pairs will repeat$)$. The prize is given to those who win both matches. Find:

a) the smallest possible number of tournament winners;

b) the mathematical expectation of the number of tournament winners.

Gary drew an empty table of $50 \times 50$ and wrote on top of each column and to the left of each row a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down a product of numbers written at the top of its column and to the left of the row $($ the “multiplication table” $)$. What is the largest number of products in this table which could be rational numbers?

Fred chose 2017 $($not necessarily different$)$ natural numbers $a_1, a_2, …, a_{2017}$ and plays by himself in the following game. Initially, he has an unlimited supply of stones and 2017 large empty boxes. In one move Fred adds a1 stones to any box $($at his choice$)$, in any of the remaining boxes $($of his choice$)$ – $a_2$ stones, …, finally, in the remaining box – $a_{2017}$ stones. His purpose is to ensure that eventually all the boxes have an equal number of stones. Could he have chosen the numbers so that the goal could be achieved in 43 moves, but is impossible for a smaller non-zero number of moves?