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

Tags were heavily modified to better represent problems.

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Found problems: 109

2010 Balkan MO Shortlist, G2
Tags: geometry , circles , areas , geometric inequality , cyclic quadrilateral , midpoints , Cyclic
Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle

2006 Oral Moscow Geometry Olympiad, 1
Tags: equal circles , construction , geometry , Cyclic
An arbitrary triangle $ABC$ is given. Construct a line that divides it into two polygons, which have equal radii of the circumscribed circles. (L. Blinkov)

2012 Tournament of Towns, 4
Tags: geometry , Cyclic , Locus , circles , tangent circles
A quadrilateral $ABCD$ with no parallel sides is inscribed in a circle. Two circles, one passing through $A$ and $B$, and the other through $C$ and $D$, are tangent to each other at $X$. Prove that the locus of $X$ is a circle.

2006 Estonia Team Selection Test, 2
Tags: Circumcenter , geometry , projections , Concyclic , Cyclic , independent
The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.

1989 Tournament Of Towns, (228) 2
Tags: hexagon , Cyclic , geometry , areas
The hexagon $ABCDEF$ is inscribed in a circle, $AB = BC = a, CD = DE = b$, and $EF = FA = c$. Prove that the area of triangle $BDF$ equals half the area of the hexagon. (I.P. Nagel, Yevpatoria).

1988 All Soviet Union Mathematical Olympiad, 467
Tags: ratio , fixed , geometry , midpoints , Cyclic
The quadrilateral $ABCD$ is inscribed in a fixed circle. It has $AB$ parallel to $CD$ and the length $AC$ is fixed, but it is otherwise allowed to vary. If $h$ is the distance between the midpoints of $AC$ and $BD$ and $k$ is the distance between the midpoints of $AB$ and $CD$, show that the ratio $h/k$ remains constant.

2019 Tournament Of Towns, 5
Tags: tangential , cyclic quadrilateral , Cyclic , geometry , area , Kvant
The point $M$ inside a convex quadrilateral $ABCD$ is equidistant from the lines $AB$ and $CD$ and is equidistant from the lines $BC$ and $AD$. The area of $ABCD$ occurred to be equal to $MA\cdot MC +MB \cdot MD$. Prove that the quadrilateral $ABCD$ is a) tangential (circumscribed), b) cyclic (inscribed). (Nairi Sedrakyan)

Estonia Open Senior - geometry, 1994.2.2
Tags: geometry , area , Cyclic , cyclic quadrilateral , equal segments
The two sides $BC$ and $CD$ of an inscribed quadrangle $ABCD$ are of equal length. Prove that the area of this quadrangle is equal to $S =\frac12 \cdot AC^2 \cdot \sin \angle A$

2015 Bulgaria National Olympiad, 1
Tags: geometry , hexagon , Cyclic , Product
The hexagon $ABLCDK$ is inscribed and the line $LK$ intersects the segments $AD, BC, AC$ and $BD$ in points $M, N, P$ and $Q$, respectively. Prove that $NL \cdot KP \cdot MQ = KM \cdot PN \cdot LQ$.