Olympiad problems

2012 Bosnia And Herzegovina - Regional Olympiad, 4

Let $S$ be an incenter of triangle $ABC$ and let incircle touch sides $AC$ and $AB$ in points $P$ and $Q$, respectively. Lines $BS$ and $CS$ intersect line $PQ$ in points $M$ and $N$, respectively. Prove that points $M$, $N$, $B$ and $C$ are concyclic

2022 European Mathematical Cup, 3

Tags: geometry , incircle , parallel lines , angle bisector

Let $ABC$ be an acute-angled triangle with $AC > BC$, with incircle $\tau$ centered at $I$ which touches $BC$ and $AC$ at points $D$ and $E$, respectively. The point $M$ on $\tau$ is such that $BM \parallel DE$ and $M$ and $B$ lie on the same halfplane with respect to the angle bisector of $\angle ACB$. Let $F$ and $H$ be the intersections of $\tau$ with $BM$ and $CM$ different from $M$, respectively. Let $J$ be a point on the line $AC$ such that $JM \parallel EH$. Let $K$ be the intersection of $JF$ and $\tau$ different from $F$. Prove that $ME \parallel KH$.

1983 Czech and Slovak Olympiad III A, 6

Tags: geometry , locus of points , incircle , triangle

Consider a circle $k$ with center $S$ and radius $r$. Denote $\mathsf M$ the set of all triangles with incircle $k$ such that the largest inner angle is twice bigger than the smallest one. For a triangle $\mathcal T\in\mathsf M$ denote its vertices $A,B,C$ in way that $SA\ge SB\ge SC$. Find the locus of points $\{B\mid\mathcal T\in\mathsf M\}$.

1967 Vietnam National Olympiad, 3

Tags: rhombus , octagon , incircle , construction , geometry

i) $ABCD$ is a rhombus. A tangent to the inscribed circle meets $AB, DA, BC, CD$ at $M, N, P, Q$ respectively. Find a relationship between $BM$ and $DN$. ii) $ABCD$ is a rhombus and $P$ a point inside. The circles through $P$ with centers $A, B, C, D$ meet the four sides $AB, BC, CD, DA$ in eight points. Find a property of the resulting octagon. Use it to construct a regular octagon. iii) Rotate the figure about the line $AC$ to form a solid. State a similar result.

1980 Poland - Second Round, 6

Tags: incircle , fixed , constant , geometry

Prove that if the point $ P $ runs through a circle inscribed in the triangle $ ABC $, then the value of the expression $ a \cdot PA^2 + b \cdot PB^2 + c \cdot PC^2 $ is constant ($ a, b, c $ are the lengths of the sides opposite the vertices $ A, B, C $, respectively).

2019 Yasinsky Geometry Olympiad, p2

Tags: geometry , isosceles , centroid , incircle

An isosceles triangle $ABC$ ($AB = AC$) with an incircle of radius $r$ is given. We know that the point $M$ of the intersection of the medians of the triangle $ABC$ lies on this circle. Find the distance from the vertex $A$ to the point of intersection of the bisectrix of the triangle $ABC$. (Grigory Filippovsky)

Geometry Mathley 2011-12, 4.3

Tags: geometry , circles , incircle , circumcircle , concurrency , tangent circle

Let $ABC$ be a triangle not being isosceles at $A$. Let $(O)$ and $(I)$ denote the circumcircle and incircle of the triangle. $(I)$ touches $AC$ and $AB$ at $E, F$ respectively. Points $M$ and $N$ are on the circle $(I)$ such that $EM \parallel FN \parallel BC$. Let $P,Q$ be the intersections of $BM,CN$ and $(I)$. Prove that i) $BC,EP, FQ$ are concurrent, and denote by $K$ the point of concurrency. ii) the circumcircles of triangle $BPK, CQK$ are all tangent to $(I)$ and all pass through a common point on the circle $(O)$. Nguyễn Minh Hà

2023 Brazil EGMO Team Selection Test, 3

Tags: geometry , circumcircle , circumcenter , incenter , incircle , concurrency

Let $\Delta ABC$ be a triangle and $L$ be the foot of the bisector of $\angle A$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle ABL$ and $\triangle ACL$ respectively and let $B_1$ and $C_1$ be the projections of $C$ and $B$ through the bisectors of the angles $\angle B$ and $\angle C$ respectively. The incircle of $\Delta ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively and the bisectors of angles $\angle B$ and $\angle C$ meet the perpendicular bisector of $AL$ at points $Q$ and $P$ respectively. Prove that the five lines $PC_0, QB_0, O_1C_1, O_2B_1$ and $BC$ are all concurrent.

1973 IMO Shortlist, 5

Tags: geometry , incircle , locus , 3d geometry , locus problems

A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.

2007 IMO Shortlist, 8

Tags: geometry , quadrilateral , incircle , triangle

Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear. [i]Author: Waldemar Pompe, Poland[/i]

2016 Saint Petersburg Mathematical Olympiad, 6

Tags: geometry , incircle , incenter , petersburg

Incircle of $\triangle ABC$ touch $AC$ at $D$. $BD$ intersect incircle at $E$. Points $F,G$ on incircle are such points, that $FE \parallel BC,GE \parallel AB$. $I_1,I_2$ are incenters of $DEF,DEG$. Prove that angle bisector of $\angle GDF$ passes though the midpoint of $I_1I_2 $.

Russian TST 2016, P3

Tags: geometry , incircle

The scalene triangle $ABC$ has incenter $I{}$ and circumcenter $O{}$. The points $B_A$ and $C_A$ are the projections of the points $B{}$ and $C{}$ onto the line $AI$. A circle with a diameter $B_AC_A$ intersects the line $BC$ at the points $K_A$ and $L_A$. [list=i] [*]Prove that the circumcircle of the triangle $AK_AL_A$ touches the incircle of the triangle $ABC$ at some point $T_A$. [*]Define the points $T_B$ and $T_C$ analogously. Prove that the lines $AT_A,BT_B$ and $CT_C$ intersect on the line $OI$. [/list]

2015 Bosnia Herzegovina Team Selection Test, 6

Tags: geometry , incircle , incenter , perpendicular bisector , parallel

Let $D$, $E$ and $F$ be points in which incircle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$, respectively, and let $I$ be a center of that circle.Furthermore, let $P$ be a foot of perpendicular from point $I$ to line $AD$, and let $M$ be midpoint of $DE$. If $\{N\}=PM\cap{AC}$, prove that $DN \parallel EF$

Russian TST 2014, P1

Tags: incircle , geometry

The inscribed circle of the triangle $ABC{}$ touches the sides $BC,CA$ and $AB{}$ at $A',B'$ and $C'{}$ respectively. Let $I_a$ be the $A$-excenter of $ABC{}.$ Prove that $I_aA'$ is perpendicular to the line determined by the circumcenters of $I_aBC'$ and $I_aCB'.$

2019 India PRMO, 28

Tags: incircle , incenter , geometry

Let $ABC$ be a triangle with sides $51, 52, 53$. Let $\Omega$ denote the incircle of $\bigtriangleup ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed $r_1 + r_2 + r_3$.

2010 Argentina National Olympiad, 2

Tags: geometry , incircle , right triangle

Let $ABC$ be a triangle with $\angle C = 90^o$ and $AC = 1$. The median $AM$ intersects the incircle at the points $P$ and $Q$, with $P$ between $A$ and $Q$, such that $AP = QM$. Find the length of $PQ$.

Kharkiv City MO Seniors - geometry, 2021.10.5

Tags: collinear , incircle , geometry

The inscribed circle $\Omega$ of triangle $ABC$ touches the sides $AB$ and $AC$ at points $K$ and $ L$, respectively. The line $BL$ intersects the circle $\Omega$ for the second time at the point $M$. The circle $\omega$ passes through the point $M$ and is tangent to the lines $AB$ and $BC$ at the points $P$ and $Q$, respectively. Let $N$ be the second intersection point of circles $\omega$ and $\Omega$, which is different from $M$. Prove that if $KM \parallel AC$ then the points $P, N$ and $L$ lie on one line.

Indonesia MO Shortlist - geometry, g2

Tags: geometry , incircle , right angle

Given an acute triangle $ABC$. The inscribed circle of triangle $ABC$ is tangent to $AB$ and $AC$ at $X$ and $Y$ respectively. Let $CH$ be the altitude. The perpendicular bisector of the segment $CH$ intersects the line $XY$ at $Z$. Prove that $\angle BZC = 90^o.$

Ukrainian TYM Qualifying - geometry, IV.10

Tags: geometry , incircle , geometric inequality

Given a triangle $ABC$ and points $D, E, F$, which are points of contact of the inscribed circle to the sides of the triangle. i) Prove that $\frac{2pr}{R} \le DE + EF + DF \le p$ ($p$ is the semiperimeter, $r$ and $R$ are respectively the radius of the inscribed and circumscribed circle of $\vartriangle ABC$). ii). Find out when equality is achieved.

2024 Bangladesh Mathematical Olympiad, P5

Tags: geometry , incircle , homothety , angle chasing , excircle

Let $I$ be the incenter of $\triangle ABC$ and $P$ be a point such that $PI$ is perpendicular to $BC$ and $PA$ is parallel to $BC$. Let the line parallel to $BC$, which is tangent to the incircle of $\triangle ABC$, intersect $AB$ and $AC$ at points $Q$ and $R$ respectively. Prove that $\angle BPQ = \angle CPR$.