Olympiad problems

2012 Sharygin Geometry Olympiad, 5

A quadrilateral $ABCD$ with perpendicular diagonals is inscribed into a circle $\omega$. Two arcs $\alpha$ and $\beta$ with diameters AB and $CD$ lie outside $\omega$. Consider two crescents formed by the circle $\omega$ and the arcs $\alpha$ and $\beta$ (see Figure). Prove that the maximal radii of the circles inscribed into these crescents are equal. (F.Nilov)

2018 All-Russian Olympiad, 2

Tags: inscribed circle , geometry , homothety

Circle $\omega$ is tangent to sides $AB, AC$ of triangle $ABC$. A circle $\Omega$ touches the side $AC$ and line $AB$ (produced beyond $B$), and touches $\omega$ at a point $L$ on side $BC$. Line $AL$ meets $\omega, \Omega$ again at $K, M$. It turned out that $KB \parallel CM$. Prove that $\triangle LCM$ is isosceles.

2004 Spain Mathematical Olympiad, Problem 5

Tags: median , inscribed circle , geometry

Demonstrate that the condition necessary so that, in triangle ${ABC}$, the median from ${B}$ is divided into three equal parts by the inscribed circumference of a circle is: ${A/5 = B/10 = C/13}$.

2018 Peru IMO TST, 3

Tags: geometry , pentagon , inscribed circle

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2018 Germany Team Selection Test, 2

Tags: inscribed circle , pentagon , geometry

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2011 IFYM, Sozopol, 5

Tags: quadrilateral , inscribed circle , geometry

A circle is inscribed in a quadrilateral $ABCD$, which is tangent to its sides $AB$, $BC$, $CD$, and $DC$ in points $M$, $N$, $P$, and $Q$ respectively. Prove that the lines $MP$, $NQ$, $AC$, and $BD$ intersect in one point.

2018 Brazil Team Selection Test, 1

Tags: inscribed circle , pentagon , geometry

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2018 Germany Team Selection Test, 2

Tags: geometry , inscribed circle , pentagon

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2021 Brazil National Olympiad, 6

Tags: bicentric polygon , convex polygon , inscribed circle , circumscribed , tangent circle , geometry

Let $n \geq 5$ be integer. The convex polygon $P = A_{1} A_{2} \ldots A_{n}$ is bicentric, that is, it has an inscribed and circumscribed circle. Set $A_{i+n}=A_{i}$ to every integer $i$ (that is, all indices are taken modulo $n$). Suppose that for all $i, 1 \leq i \leq n$, the rays $A_{i-1} A_{i}$ and $A_{i+2} A_{i+1}$ meet at the point $B_{i}$. Let $\omega_{i}$ be the circumcircle of $B_{i} A_{i} A_{i+1}$. Prove that there is a circle tangent to all $n$ circles $\omega_{i}$, $1 \leq i \leq n$.

2015 IFYM, Sozopol, 6

Tags: concurrency , geometry , inscribed circle , excircle , collinear

In $\Delta ABC$ points $A_1$, $B_1$, and $C_1$ are the tangential points of the excircles of $ABC$ with its sides. a) Prove that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $N$. b) If $AC+BC=3AB$, prove that the center of the inscribed circle of $ABC$, its tangential point with $AB$, and the point $N$ are collinear.

2025 Philippine MO, P4

Tags: parallelogram , arbitrary point , excircle , inscribed circle , circumcircle , geometry , angle bisector

Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on side $BC$. Points $X$ and $Y$ are chosen on lines $BI$ and $CI$ respectively such that $DXIY$ is a parallelogram. Points $E$ and $F$ are chosen on side $BC$ such that $AX$ and $AY$ are the angle bisectors of angles $\angle BAE$ and $\angle CAF$ respectively. Let $\omega$ be the circle tangent to segment $EF$, the extension of $AE$ past $E$, and the extension of $AF$ past $F$. Prove that $\omega$ is tangent to the circumcircle of triangle $ABC$.

2017 IMO Shortlist, G1

Tags: geometry , inscribed circle , pentagon

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2020 Junior Balkan Team Selection Tests - Moldova, 6

Tags: inscribed circle , cyclic quadrilateral , geometry

The inscribed circle inside triangle $ABC$ intersects side $AB$ in $D$. The inscribed circle inside triangle $ADC$ intersects sides $AD$ in $P$ and $AC$ in $Q$.The inscribed circle inside triangle $BDC$ intersects sides $BC$ in $M$ and $BD$ in $N$. Prove that $P , Q, M, N$ are cyclic.

2018 Taiwan TST Round 1, 1

Tags: geometry , pentagon , inscribed circle

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2007 Sharygin Geometry Olympiad, 7

Tags: geometry , convex polygon , inscribed circle , regular polygon

A convex polygon is circumscribed around a circle. Points of contact of its sides with the circle form a polygon with the same set of angles (the order of angles may differ). Is it true that the polygon is regular?

2018 Thailand TST, 1

Tags: geometry , pentagon , inscribed circle

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2013 Oral Moscow Geometry Olympiad, 2

Tags: angle bisector , geometry , equal angles , inscribed circle , common tangent

Inside the angle $AOD$, the rays $OB$ and $OC$ are drawn such that $\angle AOB = \angle COD.$ Two circles are inscribed inside the angles $\angle AOB$ and $\angle COD$ . Prove that the intersection point of the common internal tangents of these circles lies on the bisector of the angle $AOD$.

2018 Morocco TST., 4

Tags: pentagon , inscribed circle , geometry

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2011 IFYM, Sozopol, 2

Tags: circumcircle , inscribed circle , geometry

On side $AB$ of $\Delta ABC$ is chosen point $M$. A circle is tangent internally to the circumcircle of $\Delta ABC$ and segments $MB$ and $MC$ in points $P$ and $Q$ respectively. Prove that the center of the inscribed circle of $\Delta ABC$ lies on line $PQ$.

2018 Azerbaijan BMO TST, 3

Tags: geometry , pentagon , inscribed circle

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.