The cities of countries $A$ and $B$ are marked on the map, which has the form of a square with vertices at points $(0, 0)$ , $ (0, 1)$ , $(1, 1)$ , $(1, 0)$ of the plane. According to the trade agreement, country $A$ must ensure the delivery of $n$ kg of wheat to $n$ cities of country $B$, located at the points of the square with coordinates $y_1,..., y_n$, $1$ kg each city. Currently, $n$ kg of wheat are distributed among $n$ cities of country $A$, located at the points of the square with coordinates $x_1,... , x_n$, $1$ kg in each city. From each city of country $A$ to each city of the country $A$ any amount of wheat can be transported (of course, not more than $1$ kg). Transportation cost is for $t$ kg of wheat from a city with coordinates $x_i$ to a city with coordinates $y_j$ is equal to $tl_{ij}$, where $l_{ij }$is the length of the segment connecting the points $x_i$ and $y_j$. The government of country A is going to implement the optimal one (i,e. the cheapest) transportation plan. (a) Is it possible to implement the optimal transportation plan so that from each city of country $A$ to transport wheat only to one city of country $B$? (b) Will the response change if country $A$ is to deliver $n+1$ kg of wheat, in city $x_1$ is $2$ kg of wheat, and $2$ kg should be delivered to city $y_1$ (when for other cities the conditions remain the same)? [hide=original wording] Мiста країн A та B позначенi на мапi, що має вигляд квадрату з вершинами в точках (0, 0), (0, 1), (1, 1), (1, 0) площини. Згiдно торгової угоди, країна A має забез- печити доставку n кг пшеницi в n мiст країни B, що розташованi в точках квадрату з координатами y1, . . . , yn, по 1 кг в кожне мiсто. Наразi n кг пшеницi розподiленi серед n мiст країни A, що розташованi в точках квадрату з координатами x1, . . . , xn, по 1 кг в кожному мiстi. З кожного мiста країни A в кожне мiсто країни B можна перевезти довiльну кiлькiсть пшеницi (звичайно, не бiльше 1 кг). Вартiсть переве- зення t кг пшеницi з мiста з координатами xi в мiсто з координатами yj дорiвнює tlij , де lij – довжина вiдрiзку, що сполучає точки xi та yj . Уряд країни A збирається реалiзувати оптимальний (тобто найдешевший) план перевезення. 1. Чи можна реалiзувати оптимальний план перевезення таким чином, щоби з кожного мiста країни A перевозити пшеницю тiльке в одне мiсто країни B? 2. Чи змiниться вiдповiдь, якщо країна A має забезпечити доставку n + 1 кг пше- ницi, в мiстi x1 знаходиться 2 кг пшеницi, i в мiсто y1 має бути доставлено 2 кг пшеницi (щодо iнших мiст умови лишаються такими ж)?[/hide]

Let $n$ be a positive integer. Determine the smallest value of the sum $a_1b_1+a_2b_2+...+a_{2n+2}b_{2n+2}$ where $(a_1,a_2,...,a_{2n+2})$ and $(b_1,b_2,...,b_{2n+2})$ are rearrangements of the binomial coefficients $2n+1 \choose 0$, $2n+1 \choose 1$,...,$2n+1 \choose 2n+1$. Justify your answer

Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

Define $S_n=\sum_{j,k=1}^{n} \frac{1}{\sqrt{j^2+k^2}}$. Find a positive constant $C$ such that the inequality $n\le S_n \le Cn$ holds for all $n \ge 3$. (Note. The smaller $C$, the better the solution.)

Let $a$, $b$, $c$ be positive real numbers. Determine the minimum value of the following expression $$ \frac{a^2+2b^2+4c^2}{b(a+2c)}$$

Find the minimum value of $4(x^2 + y^2 + z^2 + w^2) + (xy - 7)^2 + (yz - 7)^2 + (zw - 7)^2 + (wx - 7)^2$ as $x, y, z$, and $w$ range over all real numbers.

Let $n$ be a positive integer, and let $f(n) =1^n + 2^{n-1} + 3^{n-2}+ 4^{n-3}+... + (n-1)^2 + n^1$ Find the smallest possible value of $\frac{f(n+2)}{f(n)}$ .Justify your answer.

In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$. [img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]

Let $x, y$ be positive real numbers. Find the minimum of $$x^2 + xy +\frac{y^2}{2}+\frac{2^6}{x + y}+\frac{3^4}{x^3}$$

Let $a, b$ be two real numbers that satisfy $a^3 + b^3 = 8-6ab$. Find the maximum value and the minimum value that $a + b$ can take.

Let $x, y, z, a,b, c$ are pairwise different integers from the set $\{1,2,3, 4,5,6\}$. Find the smallest possible value for expression $xyz + abc - ax - by - cz$.

The lengths, $a$ and $b$, of two sides of a triangle are known. (a) What length should the third side be, in order for the largest angle of the triangle to be of the least possible value? (b) What length should the third side be in order for the smallest angle of the triangle to be of the greatest possible value?

Find the minimum value of the expression $f(a, b, c) = (a + b)^4 + (b + c)^4 + (c + a)^4 - \frac47 (a^4 + b^4 + c^4)$, as $a, b, c$ varies over the set of all real numbers

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.