a) Given $5$ points on a plane, no three of which lie on one line. Prove that four of these points can be taken as vertices of a convex quadrilateral. b) Inside a square, consider a convex quadrilateral and inside the quadrilateral, take a point $A$. It so happens that no three of the $9$ points — the vertices of the square, of the quadrilateral and $A$ — lie on one line. Prove that $5$ of these points are vertices of a convex pentagon.

A convex pentagon $P $ is divided by all its diagonals into ten triangles and one smaller pentagon $P'$. Let $N$ be the sum of areas of five triangles adjacent to the sides of $P$ decreased by the area of $P'$. The same operations are performed with the pentagon $P'$, let $N'$ be the similar difference calculated for this pentagon. Prove that $N > N'$. (A.Belov)

We have four wooden triangles with sides $3, 4, 5$ centimeters. How many convex polygons can we make by all of these triangles? (Just draw the polygons without any proof) A convex polygon is a polygon which all of it's angles are less than $180^o$ and there isn't any hole in it. For example: [img]https://1.bp.blogspot.com/-JgvF_B-uRag/W1R4f4AXxTI/AAAAAAAAIzc/Fo3qu3pxXcoElk01RTYJYZNwj0plJaKPQCK4BGAYYCw/s640/igo%2B2015.el1.png[/img]

Examine if we can place $9$ convex $6$-angled polygons the one next to the other (with common only one side or part of her) to construct a convex $39$-angled polygon.

On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.

Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.

A convex $n$-gon has exactly three obtuse interior angles. Find all possible values of $n$.

Find all convex polyhedra which have no diagonals (that is, for which every segment connecting two vertices lies on the boundary of the polyhedron).

Let $3(2^n -1)$ points be selected in the interior of a polyhedron $P$ with volume $2^n$, where n is a positive integer. Prove that there exists a convex polyhedron $U$ with volume $1$, contained entirely inside $P$, which contains none of the selected points.

A convex polygon is divided into some triangles. Let $V$ and $E$ be respectively the set of vertices and the set of egdes of all triangles (each vertex in $V$ may be some vertex of the polygon or some point inside the polygon). The polygon is said to be [i]good [/i] if the following conditions hold: i. There are no $3$ vertices in $V$ which are collinear. ii. Each vertex in $V$ belongs to an even number of edges in $E$. Find all good polygon.

The convex set $F$ does not cover a semi-circle of radius $R$. Is it possible that two sets, congruent to $F$, cover the circle of radius $R$ ? What if $F$ is not convex? ( N . B . Vasiliev , A. G . Samosvat)

Let $C_1,C_2$, and $C_3$ be points inside a bounded convex planar set $M$. Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$. Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$, then $A\cap B \ne \emptyset$

Each diagonal of a convex pentagon is parallel to one side of the pentagon. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio.

A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.

The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.

Find all possibilities: how many acute angles can there be in a convex polygon?

(a) The point $O$ lies inside the convex polygon $A_1A_2A_3...A_n$ . Consider all the angles $A_iOA_j$ where $i, j$ are distinct natural numbers from $1$ to $n$ . Prove that at least $n- 1$ of these angles are not acute . (b) Same problem for a convex polyhedron with $n$ vertices. (V. Boltyanskiy, Moscow)

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?

The convex $n$-gon ($n\ge 5$) is cut along all its diagonals. Prove that there are at least a pair of parts with the different areas.

We are given a convex quadrilateral and point $M$ inside it . The perimeter of the quadrilateral has length $L$ while the lengths of the diagonals are $D_1$ and $D_2$. Prove that the sum of the distances from $M$ to the vertices of the quadrilateral are not greater than $L + D_1 + D_2$ . (V. Prasolov)

A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.

Find the largest possible number of intersection points of the diagonals of a convex $n$-gon.

Let $ABCDEF$ be a convex hexagon with $AB = BC, CD = DE$ and $EF = FA$. Prove that the lines through $C,E,A$ perpendicular to $BD,DF,FB$ are concurrent.

Let $P$ be a convex polygon in the plane. Show that there exists a circle containing the entire polygon $P$ and having at least three adjacent vertices of $P$ on its boundary.

Given a finite set $K_0$ of points (in the plane or space). The sequence of sets $K_1, K_2, ... , K_n, ...$ is constructed according to the rule: [i]we take all the points of $K_i$, add all the symmetric points with respect to all its points, and, thus obtain $K_{i+1}$.[/i] a) Let $K_0$ consist of two points $A$ and $B$ with the distance $1$ unit between them. For what $n$ the set $K_n$ contains the point that is $1000$ units far from $A$? b) Let $K_0$ consist of three points that are the vertices of the equilateral triangle with the unit square. Find the area of minimal convex polygon containing $K_n. K_0$ below is the set of the unit volume tetrahedron vertices. c) How many faces contain the minimal convex polyhedron containing $K_1$? d) What is the volume of the above mentioned polyhedron? e) What is the volume of the minimal convex polyhedron containing $K_n$?