Olympiad problems

2024 Brazil EGMO TST, 4

Let $ABCD$ be a cyclic quadrilateral with all distinct sides that has an inscribed circle. The incircle of $ABCD$ has center $I$ and is tangent to $AB$, $BC$, $CD$, and $DA$ at points $W$, $X$, $Y$, and $Z$, respectively. Let $K$ be the intersection of the lines $WX$ and $YZ$. Prove that $KI$ is tangent to the circumcircle of triangle $AIC$.

2024 Sharygin Geometry Olympiad, 7

Tags: bicentric quadrilateral , geometry

Restore a bicentral quadrilateral if two opposite vertices and the incenter are given.

2019 Tournament Of Towns, 3

Tags: bicentric quadrilateral , tiling , cut , combinatorial geometry , combinatorics

Prove that any triangle can be cut into $2019$ quadrilaterals such that each quadrilateral is both inscribed and circumscribed. (Nairi Sedrakyan)

Kvant 2021, M2668

Tags: bicentric quadrilateral , geometry

Two circles are given for which there is a family of quadrilaterals circumscribed around the first circle and inscribed in the second. Let's denote by $a, b, c$ and $d{}$ the consecutive lengths of the sides of one of these quadrilaterals. Prove that the sum \[\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}\]does not depend on the choice of the quadrilateral. [i]Proposed by I. Weinstein[/i]

2023 Yasinsky Geometry Olympiad, 2

Tags: bicentric quadrilateral , geometry

Quadrilateral $ABCD$ is inscribed in a circle of radius $R$, and also circumscribed around a circle of radius $r$. It is known that $\angle ADB = 45^o$. Find the area of triangle $AIB$, where point $I$ is the center of the circle inscribed in $ABCD$. (Hryhoriy Filippovskyi)

VI Soros Olympiad 1999 - 2000 (Russia), 9.5

Tags: bicentric quadrilateral , geometry

Given a circle $\omega$ and three different points $A, B, C$ on it. Using a compass and a ruler, construct a point $D$ lying on the circle $\omega$ such that a circle can be inscribed in the quadrilateral $ABCD$ (points $A$, $B$, $C$, $D$ must be located on circle $\omega$ in the indicated order).

2004 Oral Moscow Geometry Olympiad, 3

Tags: construction , bicentric quadrilateral , geometry

On the board was drawn a circle with a marked center, a quadrangle inscribed in it, and a circle inscribed in it, also with a marked center. Then they erased the quadrilateral (keeping one vertex) and the inscribed circle (keeping its center). Restore any of the erased vertices of the quadrilateral using only a ruler and no more than six lines.

2024 Yasinsky Geometry Olympiad, 4

Tags: geometry , trapezoid , bicentric quadrilateral

On side $ AB $ of an isosceles trapezoid $ ABCD $ ($ AD \parallel BC $), points $ E $ and $ F $ are chosen such that a circle can be inscribed in quadrilateral $ CDEF $. Prove that the circumcircles of triangles $ ADE $ and $ BCF $ are tangent to each other. [i]Proposed by Matthew Kurskyi[/i]

Ukrainian TYM Qualifying - geometry, VI.14

Tags: geometry , bicentric quadrilateral , geometric inequality

A quadrilateral whose perimeter is equal to $P$ is inscribed in a circle of radius $R$ and is circumscribed around a circle of radius $r$. Check whether the inequality $P\le \frac{r+\sqrt{r^2+4R^2}}{2}$ holds. Try to find the corresponding inequalities for the $n$-gon ($n \ge 5$) inscribed in a circle of radius $R$ and circumscribed around a circle of radius $r$.

Mathley 2014-15, 6

Tags: geometry , circumcircle , bicentric quadrilateral , tangent circle

A quadrilateral is called bicentric if it has both an incircle and a circumcircle. $ABCD$ is a bicentric quadrilateral with $(O)$ being its circumcircle. Let $E, F$ be the intersections of $AB$ and $CD, AD$ and $BC$ respectively. Prove that there is a circle with center $O$ tangent to all of the circumcircles of the four triangles $EAD, EBC, FAB, FCD$. Nguyen Van Linh, a student of the Vietnamese College, Ha Noi