Olympiad problems

2006 Estonia Team Selection Test, 2

Tags: projection , circumcenter , concyclic , cyclic , independent , geometry

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 $F$.

1974 Czech and Slovak Olympiad III A, 5

Tags: geometry , inequalities , geometric inequalities , geometrical inequalities , cyclic , hexagon , area

Let $ABCDEF$ be a cyclic hexagon such that \[AB=BC,\quad CD=DE,\quad EF=FA.\] Show that \[[ACE]\le[BDF]\] and determine when the equality holds. ($[XYZ]$ denotes the area of the triangle $XYZ.$)

1995 Swedish Mathematical Competition, 5

Tags: cyclic quadrilateral , geometry , angle , cyclic

On a circle with center $O$ and radius $r$ are given points $A,B,C,D$ in this order such that $AB, BC$ and $CD$ have the same length $s$ and the length of $AD$ is $s+ r$.Assume that $s < r$. Determine the angles of quadrilateral $ABCD$.

2022 Durer Math Competition Finals, 4

Tags: geometry , cyclic , concurrency

$ABCD$ is a cyclic quadrilateral whose diagonals are perpendicular to each other. Let $O$ denote the centre of its circumcircle and $E$ the intersection of the diagonals. $J$ and $K$ denote the perpendicular projections of $E$ on the sides $AB$ and $BC$ . Let $F , G$ and $H$ be the midpoint line segments. Show that lines $GJ$ , $FB$ and $HK$ either pass through the same point or are parallel to each other.

2013 Saudi Arabia BMO TST, 1

Tags: geometry , cyclic

$ABCD$ is a cyclic quadrilateral such that $AB = BC = CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE = 19$ and $ED = 6$, find the possible values of $AD$.

Croatia MO (HMO) - geometry, 2023.3

Tags: hexagon , cyclic , geometry

A convex hexagon $ABCDEF$ is given, with each two opposite sides of different lengths and parallel ($AB \parallel DE$, $BC \parallel EF$ and $CD \parallel FA$). If $|AE| = |BD|$ and $|BF| = |CE|$, prove that the hexagon $ABCDEF$ is cyclic.

Croatia MO (HMO) - geometry, 2019.7

Tags: cyclic quadrilateral , ratio , geometry , cyclic , bisects segment

On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?

1999 German National Olympiad, 3

Tags: area , geometry , rectangle , cyclic

A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area $A$ of a quadrilateral with consecutive sides $a,b,c,d$: $A =\frac{a+c}{2}\frac{b+d}{2}$ (1) and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$ (2) where $p = (a+b+c+d)/2$. However, these formulas are not valid for all convex quadrilaterals. Prove that (1) holds if and only if the quadrilateral is a rectangle, while (2) holds if and only if the quadrilateral is cyclic.