Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.

$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.

If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right. \] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.

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.

Suppose that integers are given $m <n $. Consider a spreadsheet of size $n \times n $, whose cells arbitrarily record all integers from $1 $ to ${{n} ^ {2}} $. Each row of the table is colored in yellow $m$ the largest elements. Similarly, the blue colors the $m$ of the largest elements in each column. Find the smallest number of cells that are colored yellow and blue at a time

Find the smallest real constant $p$ for which the inequality holds $\sqrt{ab}- \frac{2ab}{a + b} \le p \left( \frac{a + b}{2} -\sqrt{ab}\right)$ with any positive real numbers $a, b$.

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]

Define $g(x)$ as the largest value of$ |y^2 - xy|$ for $y$ in $[0, 1]$. Find the minimum value of $g$ (for real $x$).

Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.

Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.

Let a triangle $ABC$ be given. For any point $X$ of the triangle denote $m(X)=\min\{XA,XB,XC\}.$ Find all points $X$ (of triangle $ABC$) such that $m(X)$ is maximal.

Find the smallest possible value of the expression $$\frac{(a+b) (b + c)}{a + 2b+c}$$ where $a, b, c$ are arbitrary numbers from the interval $[1,2]$.

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

For which positive $x$ does the expression $x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$ attain the smallest value?

Positive numbers $a_1,a_2,...,a_n, b_1, b_2,...,b_n$ satisfy the condition $a_1+a_2+...+a_n=b_1+ b_2+...+b_n=1$. Find the smallest possible value of the sum $$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+...+\frac{a_n^2}{a_n+b_n}$$ (V.Kolbun)

On the midline of the isosceles trapezoid $ABCD$ ($BC \parallel AD$) find the point $K$, for which the sum of the angles $\angle DAK + \angle BCK$ will be the smallest.

Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$ if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$.

Represent $1957$ as the sum of $12$ positive integer summands $a_1, a_2, ... , a_{12}$ for which the number $a_1! \cdot a_2! \cdot a_3! \cdot ... \cdot a_{12}!$ is minimal.

If $x \in R-\{-7\}$, determine the smallest value of the expression $$\frac{2x^2 + 98}{(x + 7)^2}$$

Find the largest possible value in the real numbers of the term $$\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \ne 0$.

Find the maximum and the minimum values of $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ for real numbers $x_1, x_2, y_1,y_2$ with $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.

Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.

Mummy-trolley huts are located on a straight line at points with coordinates $x_1, x_2,...., x_n$. In this village are going to build $3$ stores $A, B$ and $C$, of which will be brought every day to all Moomin-trolls chocolates, bread and water. For the delivery of chocolate, the store takes the distance from the store to the hut, raised to the square; for bread delivery , take the distance from the store to the hut; for water delivery take distance $1$, if the distance is greater than $1$ km, but do not take anything otherwise. a) Where to build each of the stores so that the total cost of all Moomin-trolls for delivery wasthe smallest? b) Where to place the TV tower, if the fee for each Moomin-troll is the maximum distance from the TV tower to the farthest hut from it? c) How will the answer change if the Moomin-troll huts are not located in a straight line, and on the plane? [hide=original wording] На прямiй розташованi хатинки Мумi-тролей в точках з координатами x1, x2, . . . , xn. В цьому селi бираються побудувати 3 магазина A, B та C, з яких будуть кожен день привозити всiм Мумi-тролям шоколадки, хлiб та воду. За доставку шоколадки мага- зин бере вiдстань вiд магазину до хатинки, пiднесену до квадрату; за доставку хлiба – вiдстань вiд магазину до хатинки; за доставку води беруть 1, якщо вiдстань бiльша 1 км, та нiчого не беруть в супротивному випадку. 1. Де побудувати кожний з магазинiв, щоб загальнi витрати всiх Мумi-тролей на доставку були найменшими? 2. Де розташувати телевежу, якщо плата для кожного Мумi-троля – максимальна вiдстань вiд телевежi до самої вiддаленої вiд неї хатинки? 3. Як змiниться вiдповiдь, якщо хатинки Мумi-тролей розташованi не на прямiй, а на площинi?[/hide]

Given a circle $\gamma$ and a point $P$ inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of $\gamma$ passing through $P$.

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 $c_2$.