Found problems: 1389
1998 China National Olympiad, 1
Let $ABC$ be a non-obtuse triangle satisfying $AB>AC$ and $\angle B=45^{\circ}$. The circumcentre $O$ and incentre $I$ of triangle $ABC$ are such that $\sqrt{2}\ OI=AB-AC$. Find the value of $\sin A$.
Cono Sur Shortlist - geometry, 2012.G5
Let $ABC$ be an acute triangle, and let $H_A$, $H_B$, and $H_C$ be the feet of the altitudes relative to vertices $A$, $B$, and $C$, respectively. Define $I_A$, $I_B$, and $I_C$ as the incenters of triangles $AH_B H_C$, $BH_C H_A$, and $CH_A H_B$, respectively. Let $T_A$, $T_B$, and $T_C$ be the intersection of the incircle of triangle $ABC$ with $BC$, $CA$, and $AB$, respectively. Prove that the triangles $I_A I_B I_C$ and $T_A T_B T_C$ are congruent.
2014 Sharygin Geometry Olympiad, 18
Let $I$ be the incenter of a circumscribed quadrilateral $ABCD$. The tangents to circle $AIC$ at points $A, C$ meet at point $X$. The tangents to circle $BID$ at points $B, D$ meet at point $Y$ . Prove that $X, I, Y$ are collinear.
2007 Baltic Way, 11
In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements:
i) $P$ is the circumcentre of triangle $ABC$.
ii) All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$.
iii) The oriented segment $PQ$ has the same direction as the oriented segment $AD$. Similarly, $QR$ has the same direction as $BE$, and $Rs$ has the same direction as $CF$.
Prove that $S$ is the incentre of triangle $ABC$.
2007 Korea Junior Math Olympiad, 7
Let the incircle of $\triangle ABC$ meet $BC,CA,AB$ at $J,K,L$. Let $D(\ne B, J),E(\ne C,K), F(\ne A,L)$ be points on
$BJ,CK,AL$. If the incenter of $\triangle ABC$ is the circumcenter of $\triangle DEF$ and $\angle BAC = \angle DEF$, prove that $\triangle ABC$ and $\triangle DEF$ are isosceles triangles.
2013 ELMO Problems, 4
Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$.
[i]Proposed by Evan Chen[/i]
2013 Balkan MO Shortlist, G1
In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic.
([i]Bulgaria[/i])
2024 Thailand TST, 2
Let $ABC$ be triangle with incenter $I$ . Let $AI$ intersect $BC$ at $D$. Point $P,Q$ lies inside triangle $ABC$ such that $\angle BPA + \angle CQA = 180^\circ$ and $B,Q,I,P,C$ concyclic in order . $BP$ intersect $CQ$ at $X$.
Prove that the intersection of $(ABC)$ and $(APQ)$ lies on line $XD$.
2010 Greece National Olympiad, 3
A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively.
Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic.
Babis
2003 All-Russian Olympiad Regional Round, 9.6
Let $I$ be the intersection point of the bisectors of triangle $ABC$. Let us denote by $A', B', C'$ the points symmetrical to $I$ wrt the sides triangle $ABC$. Prove that if a circle circumscribes around triangle $A'B'C'$ passes through vertex $B$, then $\angle ABC = 60^o$.
KoMaL A Problems 2022/2023, A. 840
The incircle of triangle $ABC$ touches the sides in $X$, $Y$ and $Z$. In triangle $XYZ$ the feet of the altitude from $X$ and $Y$ are $X'$ and $Y'$, respectively. Let line $X'Y'$ intersect the circumcircle of triangle $ABC$ at $P$ and $Q$. Prove that points $X$, $Y$, $P$ and $Q$ are concyclic.
Proposed by [i]László Simon[/i], Budapest
2022 JHMT HS, 10
In $\triangle JMT$, $JM=410$, $JT=49$, and $\angle{MJT}>90^\circ$. Let $I$ and $H$ be the incenter and orthocenter of $\triangle JMT$, respectively. The circumcircle of $\triangle JIH$ intersects $\overleftrightarrow{JT}$ at a point $P\neq J$, and $IH=HP$. Find $MT$.
2012 Dutch IMO TST, 1
A line, which passes through the incentre $I$ of the triangle $ABC$, meets its sides $AB$ and $BC$ at the points $M$ and $N$ respectively. The triangle $BMN$ is acute. The points $K,L$ are chosen on the side $AC$ such that $\angle ILA=\angle IMB$ and $\angle KC=\angle INB$. Prove that $AM+KL+CN=AC$.
[i]S. Berlov[/i]
Kharkiv City MO Seniors - geometry, 2019.10.5
In triangle $ABC$, point$ I$ is incenter , $I_a$ is the $A$-excenter. Let $K$ be the intersection point of the $BC$ with the external bisector of the angle $BAC$, and $E$ be the midpoint of the arc $BAC$ of the circumcircle of triangle $ABC$. Prove that $K$ is the orthocenter of triangle $II_aE$.
1981 IMO, 2
Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$.
2013 Princeton University Math Competition, 2
Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) and $I$ be the center of $\gamma$. Let $D$, $E$ and $F$ be the feet of the perpendiculars from $I$ to $BC$, $CA$, and $AB$ respectively. Let $D'$ be the point on $\gamma$ such that $DD'$ is a diameter of $\gamma$. Suppose the tangent to $\gamma$ through $D$ intersects the line $EF$ at $P$. Suppose the tangent to $\gamma$ through $D'$ intersects the line $EF$ at $Q$. Prove that $\angle PIQ + \angle DAD' = 180^{\circ}$.
2010 Kyiv Mathematical Festival, 3
Let $O$ be the circumcenter and $I$ be the incenter of triangle $ABC.$ Prove that if $AI\perp OB$ and $BI\perp OC$ then $CI\parallel OA$.
2006 Kyiv Mathematical Festival, 4
See all the problems from 5-th Kyiv math festival
[url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.
1988 IMO Shortlist, 13
In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that
\[ \frac {E}{E_1} \geq 2.
\]
2006 IberoAmerican, 1
In a scalene triangle $ABC$ with $\angle A = 90^\circ,$ the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R.$
The lines $RS$ and $BC$ intersect at $N,$ while the lines $AM$ and $SR$ intersect at $U.$
Prove that the triangle $UMN$ is isosceles.
2024 European Mathematical Cup, 3
Let $\omega$ be a semicircle with diamater $AB$. Let $M$ be the midpoint of $AB$. Let $X,Y$ be points on the same semiplane with $\omega$ with respect to the line $AB$ such that $AMXY$ is a parallelogram. Let $XM\cap \omega = C$ and $YM \cap \omega = D$. Let $I$ be the incenter of $\triangle XYM$. Let $AC \cap BD= E$ and $ME$ intersects $XY$ at $T$. Let the intersection point of $TI$ and $AB$ be $Q$ and let the perpendicular projection of $T$ onto $AB$ be $P$. Prove that $M$ is midpoint of $PQ$
2008 Sharygin Geometry Olympiad, 4
(A.Zaslavsky) Given three points $ C_0$, $ C_1$, $ C_2$ on the line $ l$. Find the locus of incenters of triangles $ ABC$ such that points $ A$, $ B$ lie on $ l$ and the feet of the median, the bisector and the altitude from $ C$ coincide with $ C_0$, $ C_1$, $ C_2$.
2020 Azerbaijan Senior NMO, 3
Let $ABC$ be a scalene triangle, and let $I$ be its incenter. A point $D$ is chosen on line $BC$, such that the circumcircle of triangle $BID$ intersects $AB$ at $E\neq B$, and the circumcircle of triangle $CID$ intersects $AC$ at $F\neq C$. Circumcircle of triangle $EDF$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. Lines $FD$ and $IC$ intersect at $Q$, and lines $ED$ and $BI$ intersect at $P$. Prove that $EN\parallel MF\parallel PQ$.
2004 IMO Shortlist, 7
For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$ passes through a point independent of $ X$.
2005 China Team Selection Test, 1
Triangle $ABC$ is inscribed in circle $\omega$. Circle $\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively. Also circle $\gamma$ is tangent to circle $\omega$ at point $S$. Let the intesection of $AS$ and $PQ$ be $T$. Prove that $\angle{BTP}=\angle{CTQ}$.