Consider $K$ a polygon in plane, such that the distance between any two vertices is not greater than $1$. Let $X$ and $Y$ be two points inside $K$. Show that there exist a point $Z$, lying on the border of K, such that $XZ + Y Z \le 1$

Let a$,b,c$ be sidelengths of a triangle and $m_a,m_b,m_c$ the medians at the corresponding sides. Prove that $$m_a\left(\frac{b}{a}-1\right)\left(\frac{c}{a}-1\right)+ m_b\left(\frac{a}{b}-1\right)\left(\frac{c}{b}-1\right) +m_c\left(\frac{a}{c}-1\right)\left(\frac{b}{c}-1\right)\geq 0.$$ (FYROM)

In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?

If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it. [hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]

Prove that in a triangle the following inequality holds: $$s\sqrt3 \ge \ell_a + \ell_b + \ell_c$$ where $\ell_a$ is the length of the angle bisector from angle $A$, and $s$ is the semiperimeter of the triangle

$(YUG 2)$ A park has the shape of a convex pentagon of area $50000\sqrt{3} m^2$. A man standing at an interior point $O$ of the park notices that he stands at a distance of at most $200 m$ from each vertex of the pentagon. Prove that he stands at a distance of at least $100 m$ from each side of the pentagon.

Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.

Let triangle $\vartriangle ABC$ have side lengths $AB = 6$, $BC = 8$, and $CA = 10$. Let $S_1$ be the largest square fitting inside of $\vartriangle ABC$ (sharing points on edges is allowed). Then, for $i \ge 2$, let $S_i$ be the largest square that fits inside of $\vartriangle ABC$ while remaining outside of all other squares $S_1$,$...$, $S_{i-1}$ (with ties broken arbitrarily). For all $i \ge 1$, let $m_i$ be the side length of $S_i$ and let $S$ be the set of all $m_i$. Let $x$ be the $2023$rd largest value in $S$. Compute $\log_2 \left( \frac{1}{x}\right).$ Submit your answer as a decimal $E$ to at most $3$ decimal places. If the correct answer is $A$, your score for this question will be $\max(0, 25 -2|A - E|)$, rounded to the nearest integer

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]

Let $P$ be a point in the plane. Prove that it is possible to draw three rays with origin in $P$ with the following property: for every circle with radius $r$ containing $P$ in its interior, if $P_1$, $P_2$ and $P_3$ are the intersection points of the three rays with the circle, then \[|PP_1|+|PP_2|+|PP_3|\leq 3r.\]

Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]

In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

Let $ABC$ be a triangle whose side lengths are opposite the angle $A,B,C$ are $a,b,c$ respectively. Prove that $$\frac{ab\sin{2C}+bc\sin{ 2A}+ca\sin{2B}}{ab+bc+ca}\leq\frac{\sqrt{3}}{2}$$. [i](nooonuii)[/i]

The medians of the sides $AC$ and $BC$ in the triangle $ABC$ are perpendicular to each other. Prove that $\frac12 <\frac{|AC|}{|BC|}<2$.

Let $ABCD$ be a rectangle and $U, V$ two points of its circumcircle. Lines $AU,CV$ intersect at $P$ and lines $BU,DV$ intersect at $Q$, distinct from $P$. Prove that $$\frac{1}{PQ^2} \ge \frac{1}{UV^2} - \frac{1}{AC^2}$$ Michel Bataille

A paper square is bent along the line $\ell$, which passes through its center, so that a non-convex hexagon is formed. Investigate the question of the circle of largest radius that can be placed in such a hexagon.

$9$ points are given in the interior of the unit square. Prove there exists a triangle of area $\le \frac18$ whose vertices are three of the points.

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

Let $ABCD$ be a square. Point $P, Q, R, S$ are chosen on the sides $AB$, $BC$, $CD$, $DA$, respectively, such that $AP + CR \ge AB \ge BQ + DS$. Prove that $$area \,\, (PQRS) \le \frac12 \,\, area \,\, (ABCD)$$ and determine all cases when equality holds.

Hiven an acute triangle $ABC$, consider the midpoints $M$ and $N$ of the sides $AB$ and $AC$, respectively. If point $S$ is variable on side $BC$, prove that $$(MB - MS)(NC - NS) \le 0$$

Let $ABC$ be a triangle inscribed in a circle and let \[ l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , \] where $m_a$,$m_b$, $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$, $M_b$, $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that \[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 \] and that equality holds iff $ABC$ is an equilateral triangle.

The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that: \[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}. \]

Prove that $a^2pq + b^2qr + c^2rp \le 0$, whenever $a, b$ and $c$ are the lengths of the sides of a triangle and $p + q + r = 0$ . ( J. Mustafaev , year 12 student, Baku)

Prove that for any nonisosceles triangle $l_1^2>\sqrt3 S>l_2^2$, where $l_1, l_2$ are the greatest and the smallest bisectors of the triangle and $S$ is its area.

Show that any two points lying inside a regular $ n\minus{}$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.$