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 $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively, \[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\] if a) $X_1, \ldots , X_n$ are the midpoints of the corressponding sides, b) $X_1, \ldots , X_n$ are the feet of the corressponding altitudes, c) $X_1, \ldots , X_n$ are arbitrary points on the corressponding lines. Modified version of [url=https://artofproblemsolving.com/community/c6h418634p2361975]IMO 2010 SL G3[/url] (it was question c)

Let $P$ be any point inside the parallelogram $ABCD$ and let $R$ be the radius of the circle through $A$, $B$, and $C$. Show that the distance from $P$ to the nearest vertex is not greater than $R$.

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?

Let $G$ be the point of intersection of the medians in the triangle $ABC$. Let us denote $A_1, B_1, C_1$ the second points of intersection of lines $AG, BG, CG$ with the circle circumscribed around the triangle. Prove that $AG + BG + CG \le A_1C + B_1C + C_1C$. (Yasinsky V.A.)

In a triangle $ABC$, the bisector of the angle $A$ intersects the excircle that is tangential to side $[BC]$ at two points $D$ and $E$ such that $D\in [AE]$. Prove that, $$ \frac{|AD|}{|AE|}\leq \frac{|BC|^2}{|DE|^2}. $$

The two sides of the triangle are equal to $1$ and $x$, and $ x \ge 1$. The values $a$ and $b$ are the largest and smallest angles of this triangle, respectively. Find the greatest value of $\cos a$ and the smallest value of $\cos b$.

Let $ABCDA'B'C'D'$ be a rectangular parallelepiped and $M,N, P$ projections of points $A, C$ and $B'$ respectively on the diagonal $BD'$. a) Prove that $BM + BN + BP = BD'$. b) Prove that $3 (AM^2 + B'P^2 + CN^2)\ge 2D'B^2$ if and only if $ABCDA'B'C'D'$ is a cube.

Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality \[AF + AH + AC \leq AB + AD + AE + AG.\] In what cases does equality hold? [i]Proposed by France.[/i]

Let $M$ be an arbitrary interior point of a tetrahedron $ABCD$, and let $S_A,S_B,S_C,S_D$ be the areas of the faces $BCD,ACD,ABD,ABC$, respectively. Prove that $$S_A\cdot MA+S_B\cdot MB+S_C\cdot MC+S_D\cdot MD\ge9V,$$where $V$ is the volume of $ABCD$. When does equality hold?

Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.

Let $A, B, C$ and $D$ be four different points lying on a common circle in this order. Assume that the line segment $AB$ is the (only) longest side of the inscribed quadrilateral $ABCD$. Prove that the inequality $AB + BD > AC + CD$ holds. [i](Proposed by Karl Czakler)[/i]

Given is a triangle $ABC$ with side lengths $a, b, c$ ($a = \overline{BC}$, $b = \overline{CA}$, $c = \overline{AB}$) and area $F$. The side $AB$ is extended beyond $A$ by a and beyond $B$ by $b$. Correspondingly, $BC$ is extended beyond $B$ and $C$ by $b$ and $c$, respectively. Eventually $CA$ is extended beyond $C$ and $A$ by $c$ and $a$, respectively. Connecting the outer endpoints of the extensions , a hexagon if formed with area $G$. Prove that $\frac{G}{F}>13$.

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

A tape with width $ d < AB $ and edges perpendicular to $ AB $ moves in the plane of the acute-angled triangle $ ABC $. At what position of the tape will it cover the largest part 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.

For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)

Let S be a sphere cirucmscribed on a regular tetrahedron with an edge length greater than 1. The sphere $ S $ is represented as the sum of four sets. Prove that one of these sets includes points $ P $, $ Q $ such that the length of the segment $ PQ $ exceeds 1.

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.

Prove that for each $\Delta ABC$ with an acute $\angle C$ the following inequality is true: $(a^2+b^2) cos(\alpha -\beta )\leq 2ab$.

Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

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.

We have an isosceles triangle $ABC$ with base $BC$. Through vertex $A$ draw a line $r$ parallel to $BC$. The points $P, Q$ are located on the perpendicular bisectors of $AB$ and $AC$ respectively, such that $PQ\perp BC$. They are points $M$ and $N$ on the line $r$ such that $\angle APM = \angle AQN = 90^o$. Prove that $$\frac{1}{AM} + \frac{1}{AN}\le \frac{2}{ AB}$$

On a semicircle centred at $O$ and with radius $1$ choose the respective points $A_1,A_2,...,A_{2n}$ , for $n \in N^*$. The lenght of the projection of the vector $\overrightarrow {u}=\overrightarrow{OA_1} +\overrightarrow{OA_2}+...+\overrightarrow{OA_{2n}}$ on the diameter is an odd integer. Show that the projection of that vector on the diameter is at least $1$.

Given a circumscribed quadrilateral $ABCD$. Prove that its inradius is smaller than the sum of the inradii of triangles $ABC$ and $ACD$.