Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.

Let $A_1A_2$ and $B_1B_2$ be the diagonals of a rectangle, and let $O$ be its center. Find and construct the set of all points $P$ that satisfy simultaneously the four inequaliies: $$A_1P > OP , \\A_2P > OP, \ \ B_1P > OP , \ \ B_2P > OP.$$

The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Company $XYZ$ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A$, $B$, and $C$. There are $1$, $5$, and $4$ workers at $A$, $B$, and $C$, respectively. Find the minimum possible total distance Company $XYZ$'s workers have to travel to get to $P$.

Let $ABC$ be a triangle with area $1$ (cm$^2$). Points $D,E$ and $F$ lie on the sides $AB, BC$ and CA, respectively. Prove that $min\{$area of $\vartriangle ADF,$ area of $\vartriangle BED,$ area of $\vartriangle CEF\} \le \frac14$ (cm$^2$).

Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$. [i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that $ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$, where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.

Given a triangle $ABC$ and a point $P$ on the side $AB$ . Let $Q$ be the intersection of the straight line $CP$ (different from $C$) with the circumcicle of the triangle. Prove the inequality $$\frac{\overline{PQ}}{\overline{CQ}} \le \left(\frac{\overline{AB}}{\overline{AC}+\overline{CB}}\right)^2$$ and that equality holds if and only if the $CP$ is bisector of the angle $ACB$. [img]https://cdn.artofproblemsolving.com/attachments/b/1/068fafd5564e77930160115a1cd409c4fdbf61.png[/img]

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]

There is an infinite set of points $S$ on the plane, and any $1\times 1$ square contains a finite number of points from the set $S$. Prove that there are two different points $A$ and $B$ from $S$ such that for any other point $X$ from $S$ the following inequalities hold: $$|XA|, |XB| \ge 0.999|AB|.$$

The coordinates of the triangle's vertices in the Cartesian system $XOY$ are integers. Prove that the diameter of the circle circumscribed by this triangle is not greater than the product of the lengths of the triangle's sides.

Given an isosceles triangle $ABC$ with a vertex at the point $B$. Based on $AC$, an arbitrary point $D $ is selected, different from the vertices $A$ and $C $. On the line $AC $ select the point $E $ outside the segment $AC$, for which $AE = CD$. Prove that the perimeter $\Delta BDE$ is larger than the perimeter $\Delta ABC$.

Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds: \[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]

Let $ABC$ be an equilateral triangle with circumcircle of center $O$ and radius $R$. Point $M$ is exterior to the triangle such that $S_bS_c = S_aS_b+S_aS_c$, where $S_a, S_b, S_c$ are the areas of triangles $MBC,MCA,MAB$ respectively. Prove that $OM \ge R$. Nguyễn Tiến Lâm

A plane intersecting a unit cube divides it into two polyhedra. It is known that for each polyhedron the distance between any two points of it does not exceeds $\frac32$ m. What can be the cross-sectional area of a cube drawn by a plane?

A cylindrical glass beaker is $8$ cm high and its circumference rim is $12$ cm wide . Inside, $3$ cm from the edge, there is a tiny drop of honey. In a point on its outer surface, belonging to the plane passing through the axis of the cylinder and for the drop of honey, and located $1$ cm from the base (or bottom) of the glass, there is a fly. What is the shortest path that the fly must travel, walking on the surface from the glass, to the drop of honey, and how long is said path? [hide=original wording]Un vaso de vidrio cil´ındrico tiene 8 cm de altura y su borde 12 cm de circunferencia. En su interior, a 3 cm del borde, hay una diminuta gota de miel. En un punto de su superficie exterior, perteneciente al plano que pasa por el eje del cilindro y por la gota de miel, y situado a 1 cm de la base (o fondo) del vaso, hay una mosca. ¿Cu´al es el camino m´as corto que la mosca debe recorrer, andando sobre la superficie del vaso, hasta la gota de miel, y qu´e longitud tiene dicho camino?[/hide]

Let $ABC$ be an acute triangle. Let $I$ be a point inside the triangle and let $D$ be a point on the line $AB$. The line through $D$ which is parallel to $AI$ intersects the line $AC$ at the point $E$, and the line through $D$ parallel to $BI$ intersects the line $BC$ in point $F$. prove that $$\frac{EF \cdot CI}{2} \ge area (\vartriangle ABC) $$

On the sides $BC, CA$ and $AB$ of triangle $ABC$, respectively, points $D, E$ and $F$ are chosen. Prove that $\frac12 (BC + CA + AB)<AD + BE + CF<\frac 32 (BC + CA + AB)$.

Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$. Prove that \[ \max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j \]

Let $a, b$ and $c$ denote the side lengths and $m_a, m_b$ and $m_c$ of the median's lengths in an arbitrary triangle. Show that $$\frac34 < \frac{m_a + m_b + m_c}{a + b + c}<1$$ Also show that there is no narrower range that for each triangle that contains the fraction $$\frac{m_a + m_b + m_c}{a + b + c}$$

Let be a tetahedron $ OABC $ with $ OA\perp OB\perp OC\perp OA. $ Show that $$ OH\le r\left( 1+\sqrt 3 \right) , $$ where $ H $ is the orthocenter of $ ABC $ and $ r $ is radius of the inscribed spere of $ OABC. $ [i]Valentin Vornicu[/i]

The circle $\omega{}$ is inscribed in the quadrilateral $ABCD$. Prove that the diameter of the circle $\omega{}$ does not exceed the length of the segment connecting the midpoints of the sides $BC$ and $AD$. [i]Proposed by O. Yuzhakov[/i]

The ellipsoid $E$ is contained in the simplex $S$, which is located in the unit ball B space $R^n$. Prove that the sum of the principal semi-axes of the ellipsoid $E$ is no more than units.

Prove that in every convex quadrilateral of area $1$, the sum of the lengths of the sides and diagonals is not smaller than $2(2+\sqrt2)$.

If $a, b$ and $c$ are the length of the sides of a triangle, show that $$\frac32 \le \frac{b + c}{b + c + 2a}+ \frac{a + c}{a + c + 2b}+ \frac{a + b}{a + b + 2c}\le \frac53.$$

Numbers $a, b, c$ are the length of the medians of some triangle. If $ab + bc + ac = 1$ prove that a) $a^2b + b^2c + c^2a > \frac13$ b) $a^2b + b^2c + c^2a > \frac12$ (I. Bliznets)