Found problems: 121
a) In a convex $13$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have?
b) In a convex $1950$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have?
How many diagonals does $11$-sided convex polygon have?
Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$
The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?
Suppose that each of the diagonals $AD,BE,CF$ divides the hexagon $ABCDEF$ into two parts of the same area and perimeter. Does the hexagon necessarily have a center of symmetry?
Consider $P$ a regular $9$-sided convex polygon with each side of length $1$. A diagonal at $P$ is any line joining two non-adjacent vertices of $P$. Calculate the difference between the lengths of the largest and smallest diagonal of $P$.
In a convex $2009$-gon, all diagonals are drawn. A line intersects the $2009$-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.
Denote $U$ as the set of $20$ diagonals of the regular polygon $P_1P_2P_3P_4P_5P_6P_7P_8$.
Find the number of sets $S$ which satisfies the following conditions.
1. $S$ is a subset of $U$.
2. If $P_iP_j \in S$ and $P_j P_k \in S$, and $i \neq k$, $P_iP_k \in S$.
Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$
The isosceles trapezoid $ABCD$ has perpendicular diagonals. The parallel to the bases through the intersection point of the diagonals intersects the non-parallel sides $[BC]$ and $[AD]$ in the points $P$, respectively $R$. The point $Q$ is symmetric of the point $P$ with respect to the midpoint of the segment $[BC]$. Prove that:
a) $QR = AD$,
b) $QR \perp AD$.
A square is divided into $25$ small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw?
(I Rubanov)
Can you draw some diagonals in a convex $2014$-gon so that they do not intersect, the whole $2014$-gon is divided into triangles and each vertex belongs to an odd number of these triangles?
Let \( n \geq 3 \) be a positive integer. In a convex polygon with \( n \) sides, all the internal bisectors of its \( n \) internal angles are drawn. Determine, as a function of \( n \), the smallest possible number of distinct lines determined by these bisectors.
Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.
Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that
the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the
quadrilateral bisects the other diagonal.
Find all possible values of integer $n > 3$ such that there is a convex $n$-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal.
Proposed by Mahdi Etesamifard
Let $ABCD$ be a convex quadrilateral, and $P$ be a point in its interior. The projections of $P$ on the sides of the quadrilateral lie on a circle with center $O$. Show that $O$ lies on the line through the midpoints of $AC$ and $BD$.
Angles are equal in a hexagon, three main diagonals are equal and the other six diagonals are also equal. Is it true that it has equal sides?
A convex quadrilateral $ABC$ is given. $A',B',C',D'$ are the orthocenters of triangles $BCD, CDA, DAB, ABC$ respectively. Prove that in the quadrilaterals $ABCP$ and $A'B'C'D'$, the corresponding diagonals share the intersection points in the same ratio.
The diagonals of the trapezoid $ABCD$ are perpendicular ($AD//BC, AD>BC$) . Point $M$ is the midpoint of the side of $AB$, the point $N$ is symmetric of the center of the circumscribed circle of the triangle $ABD$ wrt $AD$. Prove that $\angle CMN = 90^o$.
(A. Mudgal, India)
The length of each side and each non-principal diagonal of a convex hexagon does not exceed $1$. Prove that this hexagon contains a principal diagonal whose length does not exceed $\frac{2}{\sqrt3}$.
The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles
The diagonals of the square $ABCD$ intersect at $P$ and the midpoint of the side $AB$ is $E$. Segment $ED$ intersects the diagonal $AC$ at point $F$ and segment $EC$ intersects the diagonal $BD$ at $G$. Inside the quadrilateral $EFPG$, draw a circle of radius $r$ tangent to all the sides of this quadrilateral. Prove that $r = | EF | - | FP |$.
A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$?
$\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$
We are given the square $ABCD$. On sides $AB$ and $CD$ we are given points $ K$ and $L$ respectively, and on segment $KL$ we are given point $M$ . Prove that the second intersection point (i.e. the one other than $M$) of the intersection points of circles circumscribed around triangles $AKM$ and $MLC$ lies on the diagonal $AC$.
(V . N . Dubrovskiy)