Found problems: 86
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.
2013 Dutch Mathematical Olympiad, 3
The sides $BC$ and $AD$ of a quadrilateral $ABCD$ are parallel and the diagonals intersect in $O$. For this quadrilateral $|CD| =|AO|$ and $|BC| = |OD|$ hold. Furthermore $CA$ is the angular bisector of angle $BCD$. Determine the size of angle $ABC$.
[asy]
unitsize(1 cm);
pair A, B, C, D, O;
D = (0,0);
B = 3*dir(180 + 72);
C = 3*dir(180 + 72 + 36);
A = extension(D, D + (1,0), C, C + dir(180 - 36));
O = extension(A, C, B, D);
draw(A--B--C--D--cycle);
draw(B--D);
draw(A--C);
dot("$A$", A, N);
dot("$B$", B, SW);
dot("$C$", C, SE);
dot("$D$", D, N);
dot("$O$", O, E);
[/asy]
Attention: the figure is not drawn to scale.
2014 Hanoi Open Mathematics Competitions, 2
How many diagonals does $11$-sided convex polygon have?
2014 Sharygin Geometry Olympiad, 3
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)
2017 Korea National Olympiad, problem 1
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 $i \neq k$, $P_iP_k \in S$.
1995 Czech and Slovak Match, 5
The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at point $E$. Prove that the reflections of $E$ in the sides of quadrilateral $ABCD$ lie on a circle.
2011 Sharygin Geometry Olympiad, 10
The diagonals of trapezoid $ABCD$ meet at point $O$. Point $M$ of lateral side $CD$ and points $P, Q$ of bases $BC$ and $AD$ are such that segments $MP$ and $MQ$ are parallel to the diagonals of the trapezoid. Prove that line $PQ$ passes through point $O$.
2018 Sharygin Geometry Olympiad, 6
Let $ABCD$ be a circumscribed quadrilateral. Prove that the common point of the diagonals, the incenter of triangle $ABC$ and the centre of excircle of triangle $CDA$ touching the side $AC$ are collinear.
2013 Oral Moscow Geometry Olympiad, 4
Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.
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?
2006 Germany Team Selection Test, 3
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]