Found problems: 1065
1992 Baltic Way, 19
Let $ C$ be a circle in plane. Let $ C_1$ and $ C_2$ be nonintersecting circles touching $ C$ internally at points $ A$ and $ B$ respectively. Let $ t$ be a common tangent of $ C_1$ and $ C_2$ touching them at points $ D$ and $ E$ respectively, such that both $ C_1$ and $ C_2$ are on the same side of $ t$. Let $ F$ be the point of intersection of $ AD$ and $ BE$. Show that $ F$ lies on $ C$.
2005 Junior Balkan Team Selection Tests - Moldova, 5
Let $ABC$ be an acute-angled triangle, and let $F$ be the foot of its altitude from the vertex $C$. Let $M$ be the midpoint of the segment $CA$. Assume that $CF=BM$. Then the angle $MBC$ is equal to angle $FCA$ if and only if the triangle $ABC$ is equilateral.
2012 Sharygin Geometry Olympiad, 1
In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$. Prove that $\angle A = 2\angle B$ if and only if $AC = 2 MD$.
2005 Tournament of Towns, 4
$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$.
[i](5 points)[/i]
V Soros Olympiad 1998 - 99 (Russia), 10.6
In triangle $ABC$, the bisectors of the internal angles $AA_1$ , $BB_1$ and $CC_1$ are drawn ($A_1, B_1$, $C_1$ - on the sides of the triangle). It is known that $\angle AA_1C = \angle AC_1B_1$. Find $\angle BCA$.
Denmark (Mohr) - geometry, 2008.4
In triangle $ABC$ we have $AB = 2, AC = 6$ and $\angle A = 120^o$ . The bisector of angle $A$ intersects the side BC at the point $D$. Determine the length of $AD$. The answer must be given as a fraction with integer numerator and denominator.
1939 Moscow Mathematical Olympiad, 047
Prove that for any triangle the bisector lies between the median and the height drawn from the same vertex.
2004 Bosnia and Herzegovina Junior BMO TST, 5
In the isosceles triangle $ABC$ ($AC = BC$), $AB =\sqrt3$ and the altitude $CD =\sqrt2$. Let $E$ and $F$ be the midpoints of $BC$ and $DB$, respectively and $G$ be the intersection of $AE$ and $CF$. Prove that $D$ belongs to the angle bisector of $\angle AGF$.
2012 Kazakhstan National Olympiad, 2
Let $ABCD$ be an inscribed quadrilateral, in which $\angle BAD<90$. On the rays $AB$ and $AD$ are selected points $K$ and $L$, respectively, such that$ KA = KD, LA = LB$. Let $N$ - the midpoint of $AC$.Prove that if $\angle BNC=\angle DNC $,so $\angle KNL=\angle BCD $
2023 Canadian Junior Mathematical Olympiad, 2
An acute triangle is a triangle that has all angles less that $90^{\circ}$ ($90^{\circ}$ is a Right Angle). Let $ABC$ be a right-angled triangle with $\angle ACB =90^{\circ}.$ Let $CD$ be the altitude from $C$ to $AB,$ and let $E$ be the intersection of the angle bisector of $\angle ACD$ with $AD.$ Let $EF$ be the altitude from $E$ to $BC.$ Prove that the circumcircle of $BEF$ passes through the midpoint of $CE.$
1949-56 Chisinau City MO, 22
Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.
Indonesia MO Shortlist - geometry, g9
It is known that $ABCD$ is a parallelogram. The point $E$ is taken so that $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line that passes through $A$, intersects the segment $DC$ at point $F$ and intersects the extension of the line $BC$ at $G$. Given $EF = EG = EC$. Prove that $\ell$ is the bisector of the angle $\angle BAD$.
2012 Greece National Olympiad, 3
Let an acute-angled triangle $ABC$ with $AB<AC<BC$, inscribed in circle $c(O,R)$. The angle bisector $AD$ meets $c(O,R)$ at $K$. The circle $c_1(O_1,R_1)$(which passes from $A,D$ and has its center $O_1$ on $OA$) meets $AB$ at $E$ and $AC$ at $Z$. If $M,N$ are the midpoints of $ZC$ and $BE$ respectively, prove that:
[b]a)[/b]the lines $ZE,DM,KC$ are concurrent at one point $T$.
[b]b)[/b]the lines $ZE,DN,KB$ are concurrent at one point $X$.
[b]c)[/b]$OK$ is the perpendicular bisector of $TX$.
2009 Indonesia TST, 2
Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that
\[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}.
\]
2020 Macedonian NationŠ°l Olympiad, 3
Let $ABC$ be a triangle, and $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$, respectively, such that $AA_1, BB_1, CC_1$ are the internal angle bisectors of $\triangle ABC$. The circumcircle $k' = (A_1B_1C_1)$ touches the side $BC$ at $A_1$. Let $B_2$ and $C_2$, respectively, be the second intersection points of $k'$ with lines $AC$ and $AB$. Prove that $|AB| = |AC|$ or $|AC_1| = |AB_2|$.
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.
2005 Swedish Mathematical Competition, 3
In a triangle $ABC$ the bisectors of angles $A$ and $C$ meet the opposite sides at $D$ and $E$ respectively. Show that if the angle at $B$ is greater than $60^\circ$, then $AE +CD <AC$.
2000 AIME Problems, 10
A circle is inscribed in quadrilateral $ABCD,$ tangent to $\overline{AB}$ at $P$ and to $\overline{CD}$ at $Q.$ Given that $AP=19, PB=26, CQ=37,$ and $QD=23,$ find the square of the radius of the circle.
2020 Peru EGMO TST, 5
Let $AD$ be the diameter of a circle $\omega$ and $BC$ is a chord of $\omega$ which is perpendicular to $AD$. Let $M,N,P$ be points on the segments $AB,AC,BC$ respectively, such that $MP\parallel AC$ and $PN\parallel AB$. The line $MN$ cuts the line $PD$ in the point $Q$ and the angle bisector of $\angle MPN$ in the point $R$. Prove that the points $B,R,Q,C$ are concyclic.
2010 Vietnam National Olympiad, 3
In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$
such that $BC$ not is the diameter.Consider a point $A$ varies in
$(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$
respective is intersect of $BC$ and internal and external bisector
of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through
orthocenter of $\triangle ABC$
and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$.
1/Prove that $MN$ pass through a fixed point
2/Determint the place of $A$ such that $S_{AMN}$ has maxium value
2011 China Western Mathematical Olympiad, 4
In a circle $\Gamma_{1}$, centered at $O$, $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$. A circle $\Gamma_{2}$, centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$, and also tangent to $AB$ at $G$ and $CD$ at $H$. A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$. The line $EF$ intersects $l$ at $M$. Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$
2003 Korea - Final Round, 2
Let $M$ be the intersection of two diagonal, $AC$ and $BD$, of a rhombus $ABCD$, where angle $A<90^\circ$. Construct $O$ on segment $MC$ so that $OB<OC$ and let $t=\frac{MA}{MO}$, provided that $O \neq M$. Construct a circle that has $O$ as centre and goes through $B$ and $D$. Let the intersections between the circle and $AB$ be $B$ and $X$. Let the intersections between the circle and $BC$ be $B$ and $Y$. Let the intersections of $AC$ with $DX$ and $DY$ be $P$ and $Q$, respectively. Express $\frac{OQ}{OP}$ in terms of $t$.
2001 Canada National Olympiad, 3
Let $ABC$ be a triangle with $AC > AB$. Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$. Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$. Let $Z$ be the intersection point of $XY$ and $BC$.
Determine the value of $\frac{BZ}{ZC}$.
2006 ISI B.Stat Entrance Exam, 4
In the figure below, $E$ is the midpoint of the arc $ABEC$ and the segment $ED$ is perpendicular to the chord $BC$ at $D$. If the length of the chord $AB$ is $l_1$, and that of the segment $BD$ is $l_2$, determine the length of $DC$ in terms of $l_1, l_2$.
[asy]
unitsize(1 cm);
pair A=2dir(240),B=2dir(190),C=2dir(30),E=2dir(135),D=foot(E,B,C);
draw(circle((0,0),2)); draw(A--B--C); draw(E--D); draw(rightanglemark(C,D,E,8));
label("$A$",A,.5A); label("$B$",B,.5B); label("$C$",C,.5C); label("$E$",E,.5E); label("$D$",D,dir(-60));
[/asy]
Indonesia MO Shortlist - geometry, g8
Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are angle bisectors. Define $P_1$, $P_2$, $P_3$ respectively as the intersection point of $AD$ with $EF$, $BE$ with $DF$, $CF$ with $DE$ respectively. Prove that
$$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$