Found problems: 121
Brazil L2 Finals (OBM) - geometry, 2010.5
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 $O$.
2009 Bundeswettbewerb Mathematik, 4
How many diagonals can you draw in a convex $2009$-gon if in the finished drawing, every drawn diagonal inside the $2009$-gon may cut at most another drawn diagonal?
1998 Tournament Of Towns, 5
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)
2012 Sharygin Geometry Olympiad, 7
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)
2007 Sharygin Geometry Olympiad, 14
In a trapezium with bases $AD$ and $BC$, let $P$ and $Q$ be the middles of diagonals $AC$ and $BD$ respectively. Prove that if $\angle DAQ = \angle CAB$ then $\angle PBA = \angle DBC$.
2009 Balkan MO Shortlist, G3
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 $BD$.
1986 All Soviet Union Mathematical Olympiad, 433
Find the relation of the black part length and the white part length for the main diagonal of the
a) $100\times 99$ chess-board;
b) $101\times 99$ chess-board.
2018 Iranian Geometry Olympiad, 3
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
2023 AMC 10, 17
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 $\mathcal{P}$?
$\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$
2009 Junior Balkan Team Selection Tests - Romania, 2
Let $ABCD$ be a quadrilateral. The diagonals $AC$ and $BD$ are perpendicular at point $O$. The perpendiculars from $O$ on the sides of the quadrilateral meet $AB, BC, CD, DA$ at $M, N, P, Q$, respectively, and meet again $CD, DA, AB, BC$ at $M', N', P', Q'$, respectively. Prove that points $M, N, P, Q, M', N', P', Q'$ are concyclic.
Cosmin Pohoata
1975 All Soviet Union Mathematical Olympiad, 209
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.
1975 Bundeswettbewerb Mathematik, 3
Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.
2011 Tournament of Towns, 4
Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?
2011 Sharygin Geometry Olympiad, 1
The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles
2011 Abels Math Contest (Norwegian MO), 2a
In the quadrilateral $ABCD$ the side $AB$ has length $7, BC$ length $14, CD$ length $26$, and $DA$ length $23$. Show that the diagonals are perpendicular.
You may assume that the quadrilateral is convex (all internal angles are less than $180^o$).
1988 Tournament Of Towns, (166) 3
(a) The vertices of a regular $10$-gon are painted in turn black and white. Two people play the following game . Each in turn draws a diagonal connecting two vertices of the same colour . These diagonals must not intersect . The winner is the player who is able to make the last move. Who will win if both players adopt the best strategy?
(b) Answer the same question for the regular $12$-gon .
(V.G. Ivanov)
2012 Sharygin Geometry Olympiad, 4
Determine all integer $n > 3$ for which a regular $n$-gon can be divided into equal triangles by several (possibly intersecting) diagonals.
(B.Frenkin)
1957 Moscow Mathematical Olympiad, 346
Find all isosceles trapezoids that are divided into $2$ isosceles triangles by a diagonal.
2011 Sharygin Geometry Olympiad, 8
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?
1984 Tournament Of Towns, (070) T4
Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle.
(V. V . Proizvolov , Moscow)
2011 District Olympiad, 2
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) $QR \perp AD$.
2015 Estonia Team Selection Test, 8
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.
1995 Bundeswettbewerb Mathematik, 3
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.
2015 Sharygin Geometry Olympiad, 6
The diagonals of convex quadrilateral $ABCD$ are perpendicular. Points $A' , B' , C' , D' $ are the circumcenters of triangles $ABD, BCA, CDB, DAC$ respectively. Prove that lines $AA' , BB' , CC' , DD' $ concur.
(A. Zaslavsky)
2012 Estonia Team Selection Test, 3
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.