Olympiad problems

Durer Math Competition CD Finals - geometry, 2015.C1

Can the touchpoints of the inscribed circle of a triangle with the triangle form an obtuse triangle?

2021 Francophone Mathematical Olympiad, 3

Tags: incircle , geometry , square , tangent

Let $ABCD$ be a square with incircle $\Gamma$. Let $M$ be the midpoint of the segment $[CD]$. Let $P \neq B$ be a point on the segment $[AB]$. Let $E \neq M$ be the point on $\Gamma$ such that $(DP)$ and $(EM)$ are parallel. The lines $(CP)$ and $(AD)$ meet each other at $F$. Prove that the line $(EF)$ is tangent to $\Gamma$

2012 Oral Moscow Geometry Olympiad, 3

Tags: geometry , tangent , isotomic , midpoint , incircle

Given an equilateral triangle $ABC$ and a straight line $\ell$, passing through its center. Intersection points of this line with sides $AB$ and $BC$ are reflected wrt to the midpoints of these sides respectively. Prove that the line passing through the resulting points, touches the inscribed circle triangle $ABC$.

2016 Croatia Team Selection Test, Problem 3

Tags: circumcircle , geometry , incircle

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

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 $I_aCB'.$

2011 Sharygin Geometry Olympiad, 6

Tags: incenter , orthocenter , incircle , geometry , altitude , contact triangle

Let $BB_1$ and $CC_1$ be the altitudes of acute-angled triangle $ABC$, and $A_0$ is the midpoint of $BC$. Lines $A_0B_1$ and $A_0C_1$ meet the line passing through $A$ and parallel to $BC$ in points $P$ and $Q$. Prove that the incenter of triangle $PA_0Q$ lies on the altitude of triangle $ABC$.

2013 Czech And Slovak Olympiad IIIA, 3

Tags: incircle , parallel , geometry , parallelogram

In the parallelolgram A$BCD$ with the center $S$, let $O$ be the center of the circle of the inscribed triangle $ABD$ and let $T$ be the touch point with the diagonal $BD$. Prove that the lines $OS$ and $CT$ are parallel.

2011 Portugal MO, 5

Tags: incircle , projection , angle bisector , geometry

Let $[ABC]$ be a triangle, $D$ be the orthogonal projection of $B$ on the bisector of $\angle ACB$ and $E$ the orthogonal projection of $C$ on the bisector of $\angle ABC$ . Prove that $DE$ intersects the sides $[AB]$ and $[AC]$ at the touchpoints of the circle inscribed in the triangle $[ABC]$.

Durer Math Competition CD Finals - geometry, 2010.C5

Tags: equal segments , geometry , incircle

Let $D$ the touchpoint of the inscribed circle of triangle $ABC$ be with side $AB$ . From $A$ the perpendicular lines on the angle bisectors of vertices $B$ and $C$ intersect them at points $A_1$ and $A_2$ respectively . Prove that $A_1A_2 = AD$.

2019 Yasinsky Geometry Olympiad, p2

Tags: incircle , centroid , geometry , isosceles

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)

2015 Belarus Team Selection Test, 3

Tags: incircle , geometry , concyclic

Let the incircle of the triangle $ABC$ touch the side $AB$ at point $Q$. The incircles of the triangles $QAC$ and $QBC$ touch $AQ,AC$ and $BQ,BC$ at points $P,T$ and $D,F$ respectively. Prove that $PDFT$ is a cyclic quadrilateral. I.Gorodnin

2014 NZMOC Camp Selection Problems, 9

Tags: incircle , geometry

Let $AB$ be a line segment with midpoint $I$. A circle, centred at $I$ has diameter less than the length of the segment. A triangle $ABC$ is tangent to the circle on sides $AC$ and $BC$. On $AC$ a point $X$ is given, and on $BC$ a point $Y$ is given such that $XY$ is also tangent to the circle (in particular $X$ lies between the point of tangency of the circle with $AC$ and $C$, and similarly $Y$ lies between the point of tangency of the circle with $BC$ and $C$. Prove that $AX \cdot BY = AI \cdot BI$.

2021 Saudi Arabia Training Tests, 4

Tags: geometry , geometric inequality , incircle , angle

Let $ABC$ be a triangle with incircle $(I)$, tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. On the line $DF$, take points $M, P$ such that $CM \parallel AB$, $AP \parallel BC$. On the line $DE$, take points $N$, $Q$ such that $BN \parallel AC$, $AQ \parallel BC$. Denote $X$ as intersection of $PE$, $QF$ and $K$ as the midpoint of $BC$. Prove that if $AX = IK$ then $\angle BAC \le 60^o$.

2012 Ukraine Team Selection Test, 4

Tags: geometry , locus , incircle , isosceles

Given an isosceles triangle $ABC$ ($AB = AC$), the inscribed circle $\omega$ touches its sides $AB$ and $AC$ at points $K$ and $L$, respectively. On the extension of the side of the base $BC$, towards $B$, an arbitrary point $M$. is chosen. Line $M$ intersects $\omega$ at the point $N$ for the second time, line $BN$ intersects the second point $\omega$ at the point $P$. On the line $PK$, there is a point $X$ such that $K$ lies between $P$ and $X$ and $KX = KM$. Determine the locus of the point $X$.

2010 Mathcenter Contest, 3

Tags: geometry , incircle

Let triangle $ABC$ be a triangle right at $B$. The inscribed circle is tangent to sides $BC,CA,AB$ at points $D,E,F$, respectively. Let $CF$ intersect the circle at the point $P$. If $\angle APB=90^{\circ}$, find the value of $\dfrac{CP+CD}{PF}$. [i](tatari/nightmare)[/i]

2019 Romania Team Selection Test, 2

Tags: circumcircle , geometry , incenter , incircle , parallel

Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.

2022 Iran-Taiwan Friendly Math Competition, 3

Tags: geometry , incircle , incenter , parallel

Let $ABC$ be a scalene triangle with $I$ be its incenter. The incircle touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. $Y$, $Z$ are the midpoints of $DF$, $DE$ respectively, and $S$, $V$ are the intersections of lines $YZ$ and $BC$, $AD$, respectively. $T$ is the second intersection of $\odot(ABC)$ and $AS$. $K$ is the foot from $I$ to $AT$. Prove that $KV$ is parallel to $DT$. [i]Proposed by ltf0501[/i]

2013 BMT Spring, 14

Tags: geometry , incircle

Triangle $ABC$ has incircle $O$ that is tangent to $AC$ at $D$. Let $M$ be the midpoint of $AC$. $E$ lies on $BC$ so that line $AE$ is perpendicular to $BO$ extended. If $AC = 2013$, $AB = 2014$, $DM = 249$, find $CE$.

2000 Switzerland Team Selection Test, 13

Tags: geometry , incircle , concyclic

The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. Let $P$ be an internal point of triangle $ABC$ such that the incircle of triangle $ABP$ touches $AB$ at $D$ and the sides $AP$ and $BP$ at $Q$ and $R$. Prove that the points $E,F,R,Q$ lie on a circle.

2015 All-Russian Olympiad, 7

Tags: incircle , geometry , incenter

A scalene triangle $ABC$ is inscribed within circle $\omega$. The tangent to the circle at point $C$ intersects line $AB$ at point $D$. Let $I$ be the center of the circle inscribed within $\triangle ABC$. Lines $AI$ and $BI$ intersect the bisector of $\angle CDB$ in points $Q$ and $P$, respectively. Let $M$ be the midpoint of $QP$. Prove that $MI$ passes through the middle of arc $ACB$ of circle $\omega$.

Geometry Mathley 2011-12, 10.2

Tags: geometry , circumcircle , concurrency , incircle , radical axis , collinear

Let $ABC$ be an acute triangle, not isoceles triangle and $(O), (I)$ be its circumcircle and incircle respectively. Let $A_1$ be the the intersection of the radical axis of $(O), (I)$ and the line $BC$. Let $A_2$ be the point of tangency (not on $BC$) of the tangent from $A_1$ to $(I)$. Points $B_1,B_2,C_1,C_2$ are defined in the same manner. Prove that (a) the lines $AA_2,BB_2,CC_2$ are concurrent. (b) the radical centers circles through triangles $BCA_2, CAB_2$ and $ABC_2$ are all on the line $OI$. Lê Phúc Lữ

2009 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: geometry , incircle , right angle

In triangle $ABC$ such that $\angle ACB=90^{\circ}$, let point $H$ be foot of perpendicular from point $C$ to side $AB$. Show that sum of radiuses of incircles of $ABC$, $BCH$ and $ACH$ is $CH$

2008 Germany Team Selection Test, 2

Tags: geometry , incircle , quadrilateral , 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]

1993 Austrian-Polish Competition, 9

Tags: incircle , geometry , equilateral triangle , equilateral , inradius

Point $P$ is taken on the extension of side $AB$ of an equilateral triangle $ABC$ so that $A$ is between $B$ and $P$. Denote by $a$ the side length of triangle $ABC$, by $r_1$ the inradius of triangle $PAC$, and by $r_2$ the exradius of triangle $PBC$ opposite $P$. Find the sum $r_1+r_2$ as a function in $a$.

2022 New Zealand MO, 4

Tags: right triangle , geometry , incircle

Triangle $ABC$ is right-angled at $B$ and has incentre $I$. Points $D$, $E$ and $F$ are the points where the incircle of the triangle touches the sides $BC$, $AC$ and AB respectively. Lines $CI$ and $EF$ intersect at point $P$. Lines $DP$ and $AB$ intersect at point $Q$. Prove that $AQ = BF$.