This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 6

Denmark (Mohr) - geometry, 2007.1
Triangle $ABC$ lies in a regular decagon as shown in the figure. What is the ratio of the area of the triangle to the area of the entire decagon? Write the answer as a fraction of integers. [img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]
decagon area geometry

1984 Tournament Of Towns, (078) 3
We are given a regular decagon with all diagonals drawn. The number "$+ 1$ " is attached to each vertex and to each point where diagonals intersect (we consider only internal points of intersection). We can decide at any time to simultaneously change the sign of all such numbers along a given side or a given diagonal . Is it possible after a certain number of such operations to have changed all the signs to negative?
decagon combinatorics combinatorial geometry

Durer Math Competition CD 1st Round - geometry, 2017.C1
The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$. What is the angle between the diagonals $D_1D_3$ and $D_2D_5$?
decagon geometry angle

1968 All Soviet Union Mathematical Olympiad, 103
Given a triangle $ABC$, point $D$ on $[AB], E$ on $[AC]$, $|AD| = |DE| = |AC| , |BD| = |AE| , DE$ is parallel to $BC$. Prove that the length $|BD|$ equals to the side of a regular decagon inscribed in a circle with the radius $R=|AC|$.
decagon geometry

1908 Eotvos Mathematical Competition, 3
A regular polygon of 10 sides (a regular decagon) may be inscribed in a circle in the following two distinct ways: Divide the circumference into $10$ equal arcs and (1) join each division point to the next by straight line segments, (2) join each division point to the next but two by straight line segments. (See figures). Prove that the difference in the side lengths of these two decagons is equal to the radius of their circumscribed circle. [img]https://cdn.artofproblemsolving.com/attachments/7/9/41c38d08f4f89e07852942a493df17eaaf7498.png[/img]
decagon circumcircle geometry

2007 Denmark MO - Mohr Contest, 1
Triangle $ABC$ lies in a regular decagon as shown in the figure. What is the ratio of the area of the triangle to the area of the entire decagon? Write the answer as a fraction of integers. [img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]
decagon area geometry