A plane intersecting a unit cube divides it into two polyhedra. It is known that for each polyhedron the distance between any two points of it does not exceeds $\frac32$ m. What can be the cross-sectional area of a cube drawn by a plane?

Points $M,N$ and $P$ on the sides $BC, CA$ and $AB$ of $\vartriangle ABC$ are such that $\vartriangle MNP$ is acute. Denote by $h$ and $H$ the lengths of the shortest altitude of $\vartriangle ABC$ and the longest altitude of $\vartriangle MNP$. Prove that $h \le 2H$.

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

Two right triangles are given, of which the incircle of the first triangle is the circumcircle of the second triangle. Let the areas of the triangles be $S$ and $S'$ respectively. Prove that $\frac{S}{S'} \ge 3 +2\sqrt2$

Let $X, Y$ , and $Z$ be concentric circles with radii $1$, $13$, and $22$, respectively. Draw points $A, B$, and $C$ on $X$, $Y$ , and $Z$, respectively, such that the area of triangle $ABC$ is as large as possible. If the area of the triangle is $\Delta$, find $\Delta^2$.

Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x<y<z$). Let $$p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}$$ be its perimeter, surface area and diagonal length, respectively. Prove that $$x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$$

Prove that $d^2+(a-b)^2<c^2$ ,where $d$ is diameter of the inscribed circle of $\vartriangle ABC$

Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that [b]i.)[/b] no three points of $ S$ are collinear, and [b]ii.)[/b] for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$ Prove that: \[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n} \]

There is a point $M$ inside a circle, at a distance $OM = d$ of the center $O$. Two chords $AB$ and $CD$ are traced through $M$ that form a right angle . Join $A$ with $C$ and $B$ with $D$. Determine the cosine of the angle that must form the chord $AB$ with $OM$ so that the sum of the areas of the triangles $AMC$ and $BMD$ be minimal.

Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that \[ AB+AC\geq4DE. \]

Let $ABC$ be an equilateral triangle whose sides have length $1$. The midpoints of $AB,BC$ are $M,N$ respectively. Points $K,L$ were chosen on $AC$ so that $KLMN$ is a rectangle. Inside this rectangle are three semi-circles with the same radius, as in the picture (the endpoints are on the edges of the rectangle, and the arcs are tangent). Find the minimum possible value of the radii of the semi-circles.

In the plane, consider a point $ X$ and a polygon $ \mathcal{F}$ (which is not necessarily convex). Let $ p$ denote the perimeter of $ \mathcal{F}$, let $ d$ be the sum of the distances from the point $ X$ to the vertices of $ \mathcal{F}$, and let $ h$ be the sum of the distances from the point $ X$ to the sidelines of $ \mathcal{F}$. Prove that $ d^2 \minus{} h^2\geq\frac {p^2}{4}.$

Let $a, b, c $ be the side lengths of a right angled triangle with c > a, b. Show that $$3<\frac{c^3-a^3-b^3}{c(c-a)(c-b)}\leq \sqrt{2}+2.$$

Let $H$ a regular hexagon and let $P$ a point in the plane of $H$. Let $V(P)$ the sum of the distances from $P$ to the vertices of $H$ and let $L(P)$ the sum of the distances from $P$ to the edges of $H$. a) Find all points $P$ so that $L(P)$ is minimun b) Find all points $P$ so that $V(P)$ is minimun

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.

Given a triangle with $a,b,c$ sides and with the area $1$ ($a \ge b \ge c$). Prove that $b^2 \ge 2$.

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 $a_{1}=a_{2}= \ldots= a_{n}$.

The point $D$ is the midpoint of the side $[AB]$ of the triangle $ABC$ . The points $E$ and $F$ belong to $[AC]$ and $[BC]$ respectively. Prove that the area of triangle $DEF$ area does not exceed the sum of the areas of triangles $ADE$ and $BDF$.

The coordinates of the triangle's vertices in the Cartesian system $XOY$ are integers. Prove that the diameter of the circle circumscribed by this triangle is not greater than the product of the lengths of the triangle's sides.

One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.

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

Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality: $$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$ [I]Proposed by A. Khrabrov[/i]

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

Prove that one can construct two triangles from six edges of an arbitrary tetrahedron. (VV Proizvolov)

Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.