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]

Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$

The quadrilateral $ABCD$ has $AD = DC = CB < AB$ and $AB \parallel CD$. Points $E$ and $F$ lie on the sides $CD$ and $BC$ such that $\angle ADE = \angle AEF$. Prove that: (a) $4CF \le CB$. (b) If $4CF = CB$, then $AE$ is the angle bisector of $\angle DAF$.

Given an integer $n \ge 3$, prove that the diameter of a convex $n$-gon (interior and boundary) containing a disc of radius $r$ is (strictly) greater than $r(1 + 1/ \cos( \pi /n))$. The Editors

In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that: $$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$ For what kind of triangles does the equality hold?

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}} \]

Let $ABC$ be an acute-angled triangle and variable $D \in [BC]$ . Let's denote by $E, F$ the feet of the perpendiculars from $D$ to $AB$, $AC$ respectively . a) Show that $$\frac{4S^2}{b^2+c^2}\le DE^2 + DF^2\le max \{h_B^2 + h_C^2 \}.$$ b) Proved that, if $D_0 \in [BC]$ is the point where the minimum of the sum $DE^2 + DF^2$ is achieved, then $D_0$ is the leg of the symmetrical median of $A$ facing the bisector of angle $A$. c) Specify the position, of $D \in [BC]$ for which the maximum of the sum $DE^2 + DF^2$ is achieved. (The area of the triangle $ABC$ was denoted by $S$ and $h_b, h_c$ are the lengths of the altitudes from $B$ and $C$ respectively)

Prove that in an arbitrary convex hexagon there is a diagonal that cuts off from it a triangle whose area does not exceed $\frac16$ of the area of the hexagon. What are the properties of a convex hexagon, each diagonal of which is cut off from it is a triangle whose area is not less than $\frac16$ the area of the hexagon?

There are two identical clocks on the wall, one showing the current Moscow time and the other showing current local time. The minimum distance between the ends of their hour hands equals $m$ and the maximum distance equals $M$. Find the distance between the centres of the clocks. (S Fomin, Leningrad)

Let $a, b, c$ be the lengths of the sides of a triangle, and let $S$ be it's area. Prove that $$S \le \frac{a^2+b^2+c^2}{4\sqrt3}$$ and the equality is achieved only for an equilateral triangle.

In a half-plane, bounded by a line \(r\), equilateral triangles \(S_1, S_2, \ldots, S_n\) are placed, each with one side parallel to \(r\), and their opposite vertex is the point of the triangle farthest from \(r\). For each triangle \(S_i\), let \(T_i\) be its medial triangle. Let \(S\) be the region covered by triangles \(S_1, S_2, \ldots, S_n\), and let \(T\) be the region covered by triangles \(T_1, T_2, \ldots, T_n\). Prove that \[\text{area}(S) \leq 4 \cdot \text{area}(T).\]

Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$. Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$. Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$.

Two turtles, the leader and the slave, are crawling along the plane from point $A$ to point $B$. They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to $B$. Prove that the slave turtle crawled no more than the leading one.

A regular hexagon has positive integer side length. A laser is emitted from one of the hexagon’s corners, and is reflected off the edges of the hexagon until it hits another corner. Let $a$ be the distance that the laser travels. What is the smallest possible value of $a^2$ such that $a > 2019$? You need not simplify/compute exponents.

What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\frac12$, and no side parallel to an axis?

In triangle $ABC$, the lengths of all sides are integers, $\angle B=2 \angle A$ and $\angle C> 90^o$. Find the smallest possible perimeter of this triangle.