$(YUG 2)$ 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.

One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?

Given a circle with radius $R$ and inscribed $n$-gon with area $S$. We mark one point on every side of the given polygon. Prove that the perimeter of the polygon with the vertices in the marked points is not less than $2S/R$.

An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$

Three spheres are tangent to one plane, to a straight line perpendicular to this plane, and in pairs to each other. The radius of the largest sphere is $1$. Within what limits can the radius of the smallest sphere vary?

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

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 \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

Find inside the triangle $ABC$, points $G$ and $H$ for which, respectively, the geometric mean and the harmonic mean of the distances to the sides of the triangle acquire maximum values. In which cases is the segment $GH$ parallel to one of the sides of the triangle? Find the length of such a segment $GH$.

Given a tetrahedron $ABCD$ with edges $|AD|=|BC|= a$, $|AC|=|BD|=b$, $AB=c$ and $|CD| = d$. Determine the smallest value of the sum $|AX|+|BX|+|CX|+|DX|$, where $X$ is any point in space.

For a convex hexagon $AHSIMC$ whose side lengths are all $1$, let $Z$ and $z$ be the maximum and minimum values, respectively, of the three diagonals $AI$, $HM$, and $SC$. If $\sqrt{x}\le Z \le \sqrt{y} $ and $\sqrt{q}\le z \le \sqrt{r} $ , find the product $qrxy$, if $q$,$ r$, $x$, and $y$ are all integers.

Prove that the sum of the distances from any point in space from the vertices of a cube with edge $a$ is not less than $4\sqrt3 a$.

Find the smallest positive number $\lambda$ such that for an arbitrary $12$ points on the plane $P_1,P_2,...P_{12}$ (points may coincide), with distance between arbitrary two of them does not exceeds $1$, holds the inequality $\sum_{1\le i\le j\le 12} P_iP_j^2 \le \lambda$

Let $ABC$ be the triangle in which $AB> AC$. Circle $\omega_a$ touches the segment of the $BC$ at point $D$, the extension of the segment $AB$ towards point $B$ at the point $F$, and the extension of the segment $AC$ towards point $C$ at the point $E$. The ray $AD$ intersects circle $\omega_a$ for second time at point $M$. Denote the circle circumscribed around the triangle $CDM$ by $\omega$. Circle $\omega$ intersects the segment $DF$ at N. Prove that $FN > ND$.

a) Does there exist a triangle in which the shortest median is longer that the longest bisectrix? b) Does there exist a triangle in which the shortest bisectrix is longer that the longest altitude?

Given is a triangle $ABC$, $\angle C = 60^o$, $R$ the midpoint of side $AB$. There exist a point $P$ on the line $BC$ and a point $Q$ on the line $AC$ such that the perimeter of the triangle $PQR$ is minimal. a) Prove that and also indicate how the points $P$ and $Q$ can be constructed. b) If $AB = c$, $AC = b$, $BC = a$, then prove that the perimeter of the triangle $PQR$ equals $\frac12\sqrt{3c^2+6ab}$ .

What is the smallest value of the ratio of the lengths of the largest side of the triangle to the radius of its inscribed circle?

Let $P$ be any point inside an acute triangle. Let $D$ and $d$ be respectively the maximum and minimum distances from $P$ to any point on the perimeter of the triangle. (a) Prove that $D \ge 2d$. (b) Determine when equality holds

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 $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that \[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\] where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$

Show that in any triangle $ABC$ with $A = 90^0$ the following inequality holds: $$(AB -AC)^2(BC^2 + 4AB \cdot AC)^2 \le 2BC^6$$

The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$. Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.

The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon. (a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.) (b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter. (c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths. It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle. (d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon can be increased without changing the perimeter. (e) Prove that the polygon of maximum area must be equiangular, or have all angles equal. (f ) Prove that the polygon of maximum area is a regular polygon. PS. You had better use hide for answers.

A square is constructed on the side $AB$ of triangle $ABC$ (outside the triangle).$ O$ is the centre of the square. $M$ and $N$ are the midpoints of the sides $BC$ and $AC$. The lengths of these sides are $a$ and $b$ respectively. Find the maximal possible value of the sum $CM + ON$ (when the angle at $C$ changes). (IF Sharygin)

A circle with radius $r$ touches the sides $AB,BC,CD,DA$ of a convex quadrilateral $ABCD$ at $E,F,G,H$, respectively. The inradii of the triangles $EBF,FCG,GDH,HAE$ are equal to $r_1,r_2,r_3,r_4$. Prove that $$r_1+r_2+r_3+r_4\ge2\left(2-\sqrt2\right)r.$$

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 $1$.