Found problems: 96
Let $P$ be a convex polygon, so have all interior angles smaller than $180^o$, and let $X$ be a point in the interior of $P$. Prove that $P$ has a side $[AB]$ such that the perpendicular from $X$ to the line $AB$ lies on the side $[AB]$.
Which of the following statements are true?
(a) If a polygon can be divided into two congruent polygons by a broken line segment, it can be divided into two congruent polygons by a straight line segment.
(b) If a convex polygon can be divided into two congruent polygons by a broken line segment, it can be so divided by a straight line segment.
(c) If a convex polygon can be divided into two polygons by a broken line segment, one of which can be mapped onto the other by a combination of rotations and translations, it can be so divided by a straight line segment.
(S Markelov,)
$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.
A convex $n$-gon $P$, where $n > 3$, is dissected into equal triangles by diagonals non-intersecting inside it. Which values of $n$ are possible, if $P$ is circumscribed?
Let $ n > 2$. Show that there is a set of $ 2^{n-1}$ points in the plane, no three collinear such that no $ 2n$ form a convex $ 2n$-gon.
Show that a convex polygon with more than four sides cannot be decomposed into two others, both similar to the first (directly or inversely), by means of a single rectilinear cut. Reasonably specify which are the quadrilaterals and triangles that admit a decomposition of this type.
Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that:
\[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\]
where $ [x]$ denotes the greatest integer not exceeding $ x$.
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?
Is it true that in every convex polygon $3$ adjacent vertices can be selected such that their circumcirscribed circle can cover the entire polygon?
The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.
How many sides of the convex polygon can equal its longest diagonal?
Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE, EF = FA$.
Prove that $\frac{BC}{BE} +\frac{DE}{DA} +\frac{FA}{FC} \ge \frac{3}{2}$ . When does equality occur?
Let $P_{1}P_{2}\dots P_{2n}$ be a convex polygon with an even number of corners. Prove that there exists a diagonal $P_{i}P_{j}$ which is not parallel to any side of the polygon.
A convex $n$-gon is dissected into $m$ triangles such that each side of each triangle is either a side of another triangle or a side of the polygon. Prove that $m+n$ is even. Find the number of sides of the triangles inside the square and the number of vertices inside the square in terms of $m$ and $n$.
(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)
A convex polygon $G$ is placed inside a convex polygon $ F$ so that their boundaries have no common points. A segment $s$ joining two points on the boundary of $F$ is called a support chord for $G$ if s contains a side or only a vertex of $G$. Prove that
(a) there exists a support chord for $G$ such that its midpoint lies on the boundary of $G$,
(b) there exist at least two such chords.
(P Pushkar)
On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?
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 $K_n$?
Let \(n \geq 5\) be integer. The convex polygon \(P = A_{1} A_{2} \ldots A_{n}\) is bicentric, that is, it has an inscribed and circumscribed circle. Set \(A_{i+n}=A_{i}\) to every integer \(i\) (that is, all indices are taken modulo \(n\)). Suppose that for all \(i, 1 \leq i \leq n\), the rays \(A_{i-1} A_{i}\) and \(A_{i+2} A_{i+1}\) meet at the point \(B_{i}\). Let \(\omega_{i}\) be the circumcircle of \(B_{i} A_{i} A_{i+1}\). Prove that there is a circle tangent to all \(n\) circles \(\omega_{i}\), \(1 \leq i \leq n\).
In the interior of the convex polygon $A_1A_2...A_{2n}$ there is point $M$. Prove that at least one side of the polygon has not intersection points with the lines $MA_i$, $1\le i\le 2n$.
(Spain)