Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality \[abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.\] Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

On the sides $AB, BC$ and $AC$ of the triangle $ABC$, points $C', A'$ and $B'$ are selected, respectively, so that the angle $A'C'B'$ is right. Prove that the segment $A'B'$ is longer than the diameter of the inscribed circle of the triangle $ABC$. (M. Volchkevich)

Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$

Suppose $n>2$ and let $A_1,\dots,A_n$ be points on the plane such that no three are collinear. [b](a)[/b] Suppose $M_1,\dots,M_n$ be points on segments $A_1A_2,A_2A_3,\dots ,A_nA_1$ respectively. Prove that if $B_1,\dots,B_n$ are points in triangles $M_2A_2M_1,M_3A_3M_2,\dots ,M_1A_1M_n$ respectively then \[|B_1B_2|+|B_2B_3|+\dots+|B_nB_1| \leq |A_1A_2|+|A_2A_3|+\dots+|A_nA_1|\] Where $|XY|$ means the length of line segment between $X$ and $Y$. [b](b)[/b] If $X$, $Y$ and $Z$ are three points on the plane then by $H_{XYZ}$ we mean the half-plane that it's boundary is the exterior angle bisector of angle $\hat{XYZ}$ and doesn't contain $X$ and $Z$ ,having $Y$ crossed out. Prove that if $C_1,\dots ,C_n$ are points in ${H_{A_nA_1A_2},H_{A_1A_2A_3},\dots,H_{A_{n-1}A_nA_1}}$ then \[|A_1A_2|+|A_2A_3|+\dots +|A_nA_1| \leq |C_1C_2|+|C_2C_3|+\dots+|C_nC_1|\] Time allowed for this problem was 2 hours.

Prove that the sum of the lengths of the diagonals of an arbitrary quadrilateral is less than the sum of the lengths of its sides.

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

Medians $AD, BE$, and $CF$ of triangle $ABC$ intersect at point $M$. Is it possible that the circles with radii $MD, ME$, and $MF$ a) all have areas smaller than the area of triangle $ABC$, b) all have areas greater than the area of triangle $ABC$, c) all have areas equal to the area of triangle $ABC$?

Prove that in any acute triangle with sides $a, b, c$ circumscribed in a circle of radius $R$ the following inequality holds: $$\frac{\sqrt2}{4} <\frac{Rp}{2aR + bc} <\frac{1}{2}$$ where $p$ represents the semi-perimeter of the triangle.

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 $A_1B_1S_1$.

All sides of a convex pentagon are equal, and all angles are different. Prove that the maximum and minimum angles are adjacent to the same side of the pentagon.

Denote $T$ the Toricelli point of the triangle $ABC$. Prove that $$AB^2 \times BC^2 \times CA^2 \geq 3(TA^2\times TB + TB^2 \times TC + TC^2 \times TA)(TA\times TB^2 + TB \times TC^2 + TC \times TA^2)$$

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?

Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have \[(a+b)(b+c)(c+a)\geq 8.\] Also determine the case of equality.

Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$

Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.

Inside the convex quadrilateral $ABCD$ there is a point $M$ such that $\angle AMB = \angle ADM + \angle BCM$ and $\angle AMD = \angle ABM + \angle DCM$. Prove that $AM \cdot CM + BM \cdot DM \ge \sqrt{AB \cdot BC\cdot CD \cdot DA}$

Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality \[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\] When does equality occur?

The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.

Prove that for any interior point of a triangle the sum of squares of distances from it to the sides of a triangle is not less than $\frac{4S^2}{9R^2}$. [hide=about S,R]they are not defined, but I suppose they mean it's area and circumradii respectively[/hide]

Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\] where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively. [i]Proposed by Witold Szczechla, Poland[/i]

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 an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

An ant sits at vertex $A$ of unit square $ABCD$. He needs to get to point $C$, where the entrance to the anthill is located. Points $A$ and $C$ are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path that an ant must overcome in order to get into the anthill.

Denote $\overline{w_a}, \overline{w_b}, \overline{w_c}$ the external angle-bisectors in triangle $ABC$, prove that $$\sum \limits_{cyc} \frac{1}{w_a}\leq \sqrt{\frac{(s^2 - r^2 - 4Rr)(8R^2 - s^2 - r^2 - 2Rr)}{8s^2R^2r}}$$

Let $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.