Found problems: 121
The perimeter, that is, the sum of the lengths of all sides of a convex quadrilateral $ ABCD $, is equal to $2004$ meters; while the length of its diagonal $ AC $ is equal to $1001$ meters. Find out if the length of the other diagonal $ BD $ can:
a) To be equal to only one meter.
b) Be equal to the length of the diagonal $ AC $.
A trapezoid $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle $k$. Let $k_a$ and $k_c$ respectively be the circles with diameters $AB$ and $CD$. Compute the area of the region which is inside the circle $k$, but outside the circles $k_a$ and $k_c$.
The diagonals of an cyclic quadrilateral $ABCD$ intersect at $O$. The circumcircles of triangle $AOB$ and $COD$ intersect lines $BC$ and $AD$, for the second time, at points $M, N, P$and $Q$. Prove that the $MNPQ$ quadrilateral is inscribed in a circle of center $O$.
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 $n\ge 3$ be an integer. An [i]inner diagonal[/i] of a [i]simple $n$-gon[/i] is a diagonal that is contained in the $n$-gon. Denote by $D(P)$ the number of all inner diagonals of a simple $n$-gon $P$ and by $D(n)$ the least possible value of $D(Q)$, where $Q$ is a simple $n$-gon. Prove that no two inner diagonals of $P$ intersect (except possibly at a common endpoint) if and only if $D(P)=D(n)$.
[i]Remark:[/i] A simple $n$-gon is a non-self-intersecting polygon with $n$ vertices. A polygon is not necessarily convex.
A $2\times 2$ square was cut from a squared sheet of paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.
Is there a convex pentagon in which each diagonal is equal to a side?
Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?
The vertices of the quadrilateral $ABCD$ lie on a single circle. The diagonals of this rectangle divide the angles of the rectangle at vertices $A$ and $B$ and divides the angles at vertices $C$ and $D$ in a $1: 2$ ratio. Find angles of the quadrilateral $ABCD$.
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$. Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$, $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?
In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.
Denote the midpoints of the convex hexagon $A_1A_2A_3A_4A_5A_6$ diagonals $A_6A_2$, $A_1A_3$, $A_2A_4$, $A_3A_5$, $A_4A_6$, $A_5A_1$ as $B_1, B_2, B_3, B_4, B_5, B_6$ respectively. Prove that if the hexagon $B_1B_2B_3B_4B_5B_6$ is convex, than its area equals to the quarter of the initial hexagon.
Do there exist convex polyhedra with an arbitrary number of diagonals (a diagonal is a segment joining two vertices of a polyhedron and not lying on the surface of this polyhedron)?
(A. Blinkov)
Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$.
[i]Proposed by Alexander Ivanov, Bulgaria[/i]
How many sides of the convex polygon can equal its longest diagonal?
Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.
The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?
Which of the following is a possible number of diagonals of a convex polygon?
(A) $02$ (B) $21$ (C) $32$ (D) $54$ (E) $63$
a) Each of the side of the convex hexagon is longer than $1$. Does it necessary have a diagonal longer than $2$?
b) Each of the main diagonals of the convex hexagon is longer than $2$. Does it necessary have a side longer than $1$?
The difference between the maximal and the minimal diagonals of the regular $n$-gon equals to its side ( $n > 5$ ). Find $n$.
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals.
If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?