Found problems: 1389
2005 Taiwan TST Round 1, 2
$P$ is a point in the interior of $\triangle ABC$, and $\angle ABP = \angle PCB = 10^\circ$.
(a) If $\angle PBC = 10^\circ$ and $\angle ACP = 20^\circ$, what is the value of $\angle BAP$?
(b) If $\angle PBC = 20^\circ$ and $\angle ACP = 10^\circ$, what is the value of $\angle BAP$?
2010 Middle European Mathematical Olympiad, 9
The incircle of the triangle $ABC$ touches the sides $BC$, $CA$, and $AB$ in the points $D$, $E$ and $F$, respectively. Let $K$ be the point symmetric to $D$ with respect to the incenter. The lines $DE$ and $FK$ intersect at $S$. Prove that $AS$ is parallel to $BC$.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 5)[/i]
2014 Saint Petersburg Mathematical Olympiad, 7
$I$ - incenter , $M$- midpoint of arc $BAC$ of circumcircle, $AL$ - angle bisector of triangle $ABC$. $MI$ intersect circumcircle in $K$. Circumcircle of $AKL$ intersect $BC$ at $L$ and $P$.
Prove that $\angle AIP=90$
2021 Romanian Master of Mathematics Shortlist, G2
Let $ABC$ be a triangle with incenter $I$. The line through $I$, perpendicular to $AI$, intersects the circumcircle of $ABC$ at points $P$ and $Q$. It turns out there exists a point $T$ on the side $BC$ such that $AB + BT = AC + CT$ and $AT^2 = AB \cdot AC$. Determine all possible values of the ratio $IP/IQ$.
2021 Taiwan TST Round 3, 2
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$.
Show that $A,X,Y$ are collinear.
2011 Polish MO Finals, 2
The incircle of triangle $ABC$ is tangent to $BC,CA,AB$ at $D,E,F$ respectively. Consider the triangle formed by the line joining the midpoints of $AE,AF$, the line joining the midpoints of $BF,BD$, and the line joining the midpoints of $CD,CE$. Prove that the circumcenter of this triangle coincides with the circumcenter of triangle $ABC$.
2010 Malaysia National Olympiad, 7
Let $ABC$ be a triangle in which $AB=AC$. A point $I$ lies inside the triangle such that $\angle ABI=\angle CBI$ and $\angle BAI=\angle CAI$. Prove that \[\angle BIA=90^o+\dfrac{\angle C}{2}\]
2018 Regional Olympiad of Mexico Northeast, 2
Let $ABC$ be a triangle such that $M$ and $N$ are the midpoints of $AC$ and $BC$, respectively. Let $I$ be the incenter of $ABC$ and $E$ be the intersection of $MN$ with $Bl$. Let $P$ be a point such that $EP$ is perpendicular to $MN$ and $NP$ parallel to $IA$. Prove that $IP$ is perpendicular to $BC$.
2011 Argentina National Olympiad Level 2, 3
Let $ABC$ be a triangle of sides $AB = 15$, $AC = 14$ and $BC = 13$. Let $M$ be the midpoint of side $AB$ and let $I$ be the incenter of triangle $ABC$. The line $MI$ intersects the altitude corresponding to the side $AB$ of triangle $ABC$ at point $P$. Calculate the length of the segment $PC$.
Note: The incenter of a triangle is the intersection point of its angle bisectors.
2001 ITAMO, 5
Let $ABC$ be a triangle and $\gamma$ the circle inscribed in $ABC$. The circle $\gamma$ is tangent to side $AB$ at the point $T$. Let $D$ be the point of $\gamma$ diametrically opposite to $T$, and $S$ the intersection point of the line through $C$ and $D$ with side $AB$.
Prove that $AT=SB$.
2022 Latvia Baltic Way TST, P12
Let $\triangle ABC$ be a triangle with incentre $I$. Points $E$ and $F$ are the tangency points of the incircle and the sides $AC$ and $AB$, respectively. Suppose that the lines $BI$ and $CI$ intersect the line $EF$ at $Y$ and $Z$, respectively. Let $M$ denote the midpoint of the segment $BC$, and $N$ denote the midpoint of the segment $YZ$. Prove that $AI \parallel MN$.
2007 France Team Selection Test, 3
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
2011 China Second Round Olympiad, 11
A line $\ell$ with slope of $\frac{1}{3}$ insects the ellipse $C:\frac{x^2}{36}+\frac{y^2}{4}=1$ at points $A,B$ and the point $P\left( 3\sqrt{2} , \sqrt{2}\right)$ is above the line $\ell$.
[list]
[b](1)[/b] Prove that the locus of the incenter of triangle $PAB$ is a segment,
[b](2)[/b] If $\angle APB=\frac{\pi}{3}$, then find the area of triangle $PAB$.[/list]
2004 Regional Olympiad - Republic of Srpska, 2
Let $ABC$ be a triangle, $T$ its centroid and $S$ its incenter.
Prove that the following conditions are equivalent:
(1) line $TS$ is parallel to one side of triangle $ABC$,
(2) one of the sides of triangle $ABC$ is equal to the half-sum of the other two sides.
2023 Olympic Revenge, 5
Let $ABCD$ be a circumscribed quadrilateral and $T=AC\cap BD$. Let $I_1$, $I_2$, $I_3$, $I_4$ the incenters of $\Delta TAB$, $\Delta TBC$, $TCD$, $TDA$, respectively, and $J_1$, $J_2$, $J_3$, $J_4$ the incenters of $\Delta ABC$, $\Delta BCD$, $\Delta CDA$, $\Delta DAB$. Show that $I_1I_2I_3I_4$ is a cyclic quadrilateral and its center is $J_1J_3\cap J_2J_4$
Ukraine Correspondence MO - geometry, 2014.10
In the triangle $ABC$, it is known that $AC <AB$. Let $\ell$ be tangent to the circumcircle of triangle $ABC$ drawn at point $A$. A circle with center $A$ and radius $AC$ intersects segment $AB$ at point $D$, and line $\ell$ at points $E$ and $F$. Prove that one of the lines $DE$ and $DF$ passes through the center inscribed circle of triangle $ABC$.
2020 China Northern MO, P2
In $\triangle ABC$, $AB>AC$. Let $O$ and $I$ be the circumcenter and incenter respectively. Prove that if $\angle AIO = 30^{\circ}$, then $\angle ABC = 60^{\circ}$.
2005 Germany Team Selection Test, 3
Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality
\[9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,\]
and determine when equality holds.
2006 Germany Team Selection Test, 3
Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
1998 Belarusian National Olympiad, 6
Points $M$ and $N$ are marked on the straight line containing the side $AC$ of triangle $ABC$ so that $MA = AB$ and $NC = CB$ (the order of the points on the line: $M, A, C, N$). Prove that the center of the circle inscribed in triangle $ABC$ lies on the common chord of the circles circumscribed around triangles $MCB$ and $NAB$ .
Geometry Mathley 2011-12, 10.4
Let $A_1A_2A_3...A_n$ be a bicentric polygon with $n$ sides. Denote by $I_i$ the incenter of triangle $A_{i-1}A_iA_{i+1}, A_{i(i+1)}$ the intersection of $A_iA_{i+2}$ and $A_{i-1}A_{i+1},I_{i(i+1)}$ is the incenter of triangle $A_iA_{i(i+1)}A_{i+1}$ ($i = 1, n$). Prove that there exist $2n$ points $I_1, I_2, ..., I_n, I_{12}, I_{23}, ...., I_{n1}$ on the same circle.
Nguyễn Văn Linh
2017 IMO Shortlist, G7
A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.
1998 Romania Team Selection Test, 1
We are given an isosceles triangle $ABC$ such that $BC=a$ and $AB=BC=b$. The variable points $M\in (AC)$ and $N\in (AB)$ satisfy $a^2\cdot AM \cdot AN = b^2 \cdot BN \cdot CM$. The straight lines $BM$ and $CN$ intersect in $P$. Find the locus of the variable point $P$.
[i]Dan Branzei[/i]
2007 Iran MO (3rd Round), 3
Let $ I$ be incenter of triangle $ ABC$, $ M$ be midpoint of side $ BC$, and $ T$ be the intersection point of $ IM$ with incircle, in such a way that $ I$ is between $ M$ and $ T$. Prove that $ \angle BIM\minus{}\angle CIM\equal{}\frac{3}2(\angle B\minus{}\angle C)$, if and only if $ AT\perp BC$.
2014 Germany Team Selection Test, 2
Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$.
Prove
\[ AN \cdot NC = CD \cdot BN. \]