A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed, so that the sum of the lengths of these four cords is as small as possible?

Find the smallest value of the expression $(x-1) (x -2) (x -3) (x - 4) + 10$.

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]

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

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.

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

Given is a right triangle $ABC$ with perimeter $2$, with $\angle B=90^o$ . Point $S$ is the center of the excircle to the side $AB$ of the triangle and $H$ is the intersection of the heights of the triangle $ABS$ . Determine the smallest possible length of the segment $HS $.

Peeter, Juri, Kati and Mari are standing at the entrance of a dark tunnel. They have one torch and none of them dares to be in the tunnel without it, but the tunnel is so narrow that at most two people can move together. It takes $1$ minute for Peeter, $2$ minutes for Juri, $5$ for Kati and $10$ for Mari to pass the tunnel. Find the minimum time in which they can all pass through the tunnel.

Given positive numbers $a,b,c$, find the minimum of the function $f(x) = \sqrt{a^2 +x^2} +\sqrt{(b-x)^2 +c^2}$.

You have an equilateral paper triangle of area $9$ and fold it in two, following a straight line that passes through the center of the triangle and does not contain any vertex of the triangle. Thus there remains a quadrilateral in which the two pieces overlap, and three triangles without overlaps. Determine the smallest possible value of the quadrilateral area of the overlay.

The absolute value of the difference of the solutions of the equation $x^2 + px + q = 0$, with $p, q \in R$, is equal to $4$. Find the solutions of the equation if it is known that $(q + 1) p^2 + q^2$ takes the minimum value.

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

A cubic trinomial $x^3 + px + q$ with integer coefficients $p$ and $q$ is said to be [i]irrational [/i] if it has three pairwise distinct real irrational roots $a_1,a_2, a_3$ Find all irrational cubic trinomials for which the value of $|a_1| + [a_2| + |a_3|$ is the minimal possible. (E. Barabanov)

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.

Let $x, y, z$ be reals summing to $1$ which minimizes $2x^2 + 3y^2 + 4z^2$. Find $x$.

The positive differences $a_i-a_j$ of five different positive integers $a_1, a_2, a_3, a_4, a_5$ are all different (there are altogether $10$ such differences). Find the least possible value of the largest number among the $a_i$.

Let $a$ and $b$ be positive real numbers such that $3(a^2+b^2-1) = 4(a+b$). Find the minimum value of the expression $\frac{16}{a}+\frac{1}{b}$ .

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 $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.

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 $n$ be a positive integer. Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $n$ distinct points.

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 $xyz + abc - ax - by - cz$.

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 $[1,2]$.

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