In a triangle, each median forms an angle with the side it is drawn to, which is less than $\alpha$. Prove that one of the angles of the triangle is greater than $180^\circ-\frac{3}{2}\alpha$. [i]Proposed by Sergey Berlov[/i]

In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2 <r_a$ . (here $r$ is the radius of the circle inscribed in the triangle $ABC$, $r_a$ is the radius of an exscribed circle that touches the sides of $BC$). (Mykola Moroz)

We have $n\ge3$ points on the plane such that no three are collinear. Prove that it's possible to name them $P_1,P_2,\ldots,P_n$ such that for all $1<i<n$, the angle $\angle P_{i-1}P_iP_{i+1}$ is acute.

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

$ABC$ is a triangle with an area of $1$ square meter. Given the point $D$ on $BC$, point $E$ on $CA$, point $F$ on $AB$, such that quadrilateral $AFDE$ is cyclic. Prove that the area of $DEF \le \frac{EF^2}{4 AD^2}$. [i](holmes)[/i]

Show that the radian measure of an acute angle is less than the arithmetic mean of its sine and its tangent.

Fix the triangle $ABC$ on the plane. 1. Denote by $S_L,S_M$ and $S_K$ the areas of triangles whose vertices are, respectively, the bases of bisectors, medians and points of tangency of the inscribed circle of a given triangle $ABC$. Prove that $S_K\le S_L\le S_M$. 2. For the point $X$, which is inside the triangle $ABC$, consider the triangle $T_X$, the vertices of which are the points of intersection of the lines $AX, BX, CX$ with the lines $BC, AC, AB$, respectively. 2.1. Find the position of the point $X$ for which the area of ​​the triangle $T_x$ is the largest possible. 2.2. Suggest an effective criterion for comparing the areas of triangles $T_x$ for different positions of the point $X$. 2.3. Find the positions of the point $X$ for which the perimeter of the triangle $T_x$ is the smallest possible and the largest possible. 2.4. Propose an effective criterion for comparing the perimeters of triangles $T_x$ for different positions of point $X$. 2.5. Suggest and solve similar problems with respect to the extreme values ​​of other parameters (for example, the radius of the circumscribed circle, the length of the greatest height) of triangles $T_x$. 3. For the point $Y$, which is inside the circle $\omega$, circumscribed around the triangle $ABC$, consider the triangle $\Delta_Y$, the vertices of which are the points of intersection $AY, BX, CX$ with the circle $\omega$. Suggest and solve similar problems for triangles $\Delta_Y$ for different positions of point $Y$. 4. Suggest and solve similar problems for convex polygons. 5. For the point $Z$, which is inside the circle $\omega$, circumscribed around the triangle $ABC$, consider the triangle $F_Z$, the vertices of which are orthogonal projections of the point $Z$ on the lines $BC$, $AC$ and $AB$. Suggest and solve similar problems for triangles $F_Z$ for different positions of the point $Z$.

Given the segments $AB$ and $CD$ not necessarily on the same plane. Point $X$ is the midpoint of the segment $AB$, and the point $Y$ is the midpoint of $CD$. Given that point $X$ is not on line $CD$, and that point $Y$ is not on line $AB$, prove that $2 | XY | \le | AD | + | BC |$. When is equality achieved?

The tangents to the inscribed circle of $\triangle ABC$, which are parallel to the sides of the triangle and do not coincide with them, intersect the sides of the triangle in the points $M,N,P,Q,R,S$ such that $M,S\in (AB)$, $N,P\in (AC)$, $Q,R\in (BC)$. The interior angle bisectors of $\triangle AMN$, $\triangle BSR$ and $\triangle CPQ$, from points $A,B$ and respectively $C$ have lengths $l_{1}$ , $l_{2}$ and $l_{3}$ .\\ Prove the inequality: $\frac {1}{l^{2}_{1}}+\frac {1}{l^{2}_{2}}+\frac {1}{l^{2}_{3}} \ge \frac{81}{p^{2}}$ where $p$ is the semiperimeter of $\triangle ABC$ .

The diagonals of trapezoid $ABCD$ with bases $AB$ and $CD$ meet at $P$. Prove the inequality $S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA}$, where $S_{XYZ}$ denotes the area of triangle $XYZ$.

A tetrahedron $OA_1B_1C_1$ is given. Let $A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1$ be points such that the planes $A_1B_1C_1,A_2B_2C_2$ and $A_3B_3C_3$ are parallel and $OA_1 > OA_2 > OA_3 > 0$. Let $V_i$ be the volume of the tetrahedron $OA_iB_iC_i$ ($i = 1,2,3$) and $V$ be the volume of $OA_1B_2C_3$. Prove that $V_1 +V_2 +V_3 \ge 3V$.

Denote an acute-angle $\vartriangle ABC $ with sides $a, b, c $ respectively by ${{H}_{a}}, {{H}_{b}}, {{H}_{c}} $ the feet of altitudes ${{h}_{a}}, {{h}_{b}}, {{h}_{c}} $. Prove the inequality: $$\frac {h_ {a} ^{2}} {{{a} ^{2}} - CH_ {a} ^{2}} + \frac{h_{b} ^{2}} {{{ b}^{2}} - AH_{b} ^{2}} + \frac{h_{c}^{2}}{{{c}^{2}} - BH_{c}^{2}} \ge 3 $$ (Dmitry Petrovsky)

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: \[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]

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.$

Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.

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)

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)

Determine the set of points $P$ in the plane of a square $ABCD$ for which \[\max (PA, PC)=\frac1{\sqrt2}(PB+PD).\] [i]Titu Andreescu and I.V. Maftei[/i]

Let $ABC$ be an acute triangle, with median $AM$, height $AH$ and internal angle bisector $AL$. Suppose that $B, H, L, M, C$ are collinear in that order, and $LH<LM$. Prove that $BC>2AL$.

Given a triangle $ABC$ and a point $O$ inside it, it is known that $AB\le BC\le CA$. Prove that $$OA+OB+OC<BC+CA.$$

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

Suppose four coplanar points $A, B, C$, and $D$ satisfy $AB = 3$, $BC = 4$, $CA = 5$, and $BD = 6$. Determine the maximal possible area of $\vartriangle ACD$.

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that \[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \] [i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$. Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$. [i]Proposed by Dirk Laurie, South Africa[/i]

Two equilateral triangles are inscribed in a circle with radius $r$. Let $A$ be the area of the set consisting of all points interior to both triangles. Prove that $2A \geq r^2 \sqrt 3.$

$ABCD$ is a regular tetrahedron. The points $P,Q,R$ and $S$ lie outside this tetrahedron in such a way that $ABCP$, $ABDQ$, $ACDR$ and $BCDS$ are regular tetrahedra. Prove that the volume of the tetrahedron $PQRS$ is less than the sum of the volumes of $ABCP$,$ABDQ$,$ACDR$, $BCDS$ and $ABCD$.