Μαθηματικοί Διαγωνισμοί 1 / Γεωμετρία - Στεργίου & Μπραζιτίκος
λυμένες
3.1 - 3.2
3.3 2000 JBMO Shortlist 21
All the angles of the hexagon ${ABCDEF}$ are equal. Prove that ${AB-DE=EF-BC=CD-FA}$
3.4 2000 JBMO problem 3 (ALB)
A half-circle of diameter ${EF }$ is placed on the side ${BC }$ of a triangle ${ABC }$ and it is tangent to the sides ${AB }$ and ${AC }$ in the points ${Q }$ and ${P }$ respectively. Prove that the intersection point ${K }$ between the lines ${EP }$ and ${F Q }$ lies on the altitude from ${A }$ of the triangle ${ABC }$.
3.5 2001 JBMO Shortlist 12 (GRE)
Consider the triangle ${ABC}$ with ${\angle A = 90^\circ}$ and ${\angle B \ne \angle C}$. A circle ${C(O,R) }$ passes through ${B}$ and ${C}$ and intersects the sides ${AB}$ and ${AC}$ at ${D}$ and ${E}$, respectively. Let ${S}$ be the foot of the perpendicular from ${A}$ to ${BC}$ and let ${K}$ be the intersection point of ${AS}$ with the segment ${DE}$. If ${M}$ is the midpoint of ${BC}$, prove that ${AKOM}$ is a parallelogram.
3.6 2002 JBMO Shortlist G2 (9)
In triangle ${ABC, H,I,O}$ are orthocenter, incenter and circumcenter, respectively. ${CI}$ cuts circumcircle at ${L}$. If ${AB = IL}$ and ${AH = OH}$, find angles of triangle ${ABC}$.
3.7 2005 JBMO Shortlist G5 (9)
Let $C_1,C_2$ be two circles intersecting at points $A,P$ with centers $O,K$ respectively. Let $B,C$ be the symmetric of $A$ wrt $O,K$ in circles $C_1,C_2 $ respectively. A random line passing through $A$ intersects circles $C_1,C_2$ at $D,E$ respectively. Prove that the center of circumcircle of triangle $DEP$ lies on the circumcircle of triangle $OKP$.
Consider a convex quadrilateral ${ABCD}$ with ${AB = CD}$ and ${\angle BAC = 30^\circ}$. If ${\angle ADC = 150^\circ }$, prove that ${\angle BCA =\angle ACD}$.
3.93.10 1999 JBMO problem 4 (GRE)
Let ${ABC }$ be a triangle with ${AB = AC }$. Also, let ${D \in [BC] }$ be a point such that ${BC > BD >DC > 0 }$, and let ${C_1, C_2 }$ be the circumcircles of the triangles ${ABD }$ and ${ADC }$ respectively. Let ${BB' }$ and ${CC' }$ be diameters in the two circles, and let ${M }$ be the midpoint of ${B'C'}$. Prove that the area of the triangle ${MBC }$ is constant (i.e. it does not depend on the choice of the point ${D }$).
3.11 2005 JBMO Shortlist G4 (7)
Let $ABC$ be an isosceles triangle $(AB=AC)$ so that $\angle A< 2 \angle B$ . Let $D,Z $ points on the extension of height $AM$ so that $\angle CBD = \angle A$ and $\angle ZBA = 90^\circ$. Let $E$ the orthogonal projection of $M$ on height $BF$, and let $K$ the orthogonal projection of $Z$ on $AE$. Prove that $ \angle KDZ = \angle KDB = \angle KZB$.
In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE = 24 ^{\circ},$ $ \angle CED = 18 ^{\circ}.$
3.13 2005 JBMO Shortlist G6 (8)
Let $O$ be the center of the concentric circles $C_1,C_2$ of radii $3$ and $5$ respectively. Let $A\in C_1, B\in C_2$ and $C$ point so that triangle $ABC$ is equilateral. Find the maximum length of $ [OC] $.
3.14 All - Soviet Union 1968Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.
άλυτες
3.15 1997 JBMO problem 3 (GRE)
Let ${ABC }$ be a triangle and let ${I }$ be the incenter. Let ${N, M }$ be the midpoints of the sides ${AB }$ and ${CA }$ respectively. The lines ${BI }$ and ${CI }$ meet ${MN }$ at ${K }$ and ${L }$ respectively. Prove that ${AI + BI + CI > BC + KL }$.
3.16 1997 JBMO problem 4 (ROM)
Τα θέματα Γεωμετρίας από τις πιο πρόσφατες JBMO Shortlist Γεωμετρίας, μέτα την έκδοση του παραπάνω βιβλίου, από το 2010 μέχρι και σήμερα, βρίσκονται εδώ.3.16 1997 JBMO problem 4 (ROM)
Determine the triangle with sides ${a, b, c }$ and circumradius ${R }$ for which ${R(b + c) = a\sqrt{bc} }$.
3.17 1998 JBMO problem 2 (GRE)
3.42 2000 JBMO Shortlist 22
3.45
3.46 2001 JBMO Shortlist 11 (ROM)
3.51 2002 JBMO Shortlist G8 (13)
3.77 2005 JBMO Shortlist G1 (5)
3.78 2005 JBMO Shortlist G2 problem 2
3.79 2005 JBMO Shortlist G3 (6)
3.80 2005 JBMO Shortlist G7 (10)
3.81 2004 JBMO Shortlist G1 (7)
3.82 2004 JBMO Shortlist G3 (9)
Let ${ABC}$ be an isosceles triangle with ${AC = BC}$, let ${M}$ be the midpoint of its side ${AC}$, and let ${Z}$ be the line through ${C}$ perpendicular to ${AB}$. The circle through the points ${B, C}$, and ${M}$ intersects the line ${Z}$ at the points ${C}$ and ${Q}$. Find the radius of the circumcircle of the triangle ${ABC}$ in terms of ${m = CQ}$.
3.84 2004 JBMO Shortlist G5 (10)
Let $ABC$ be a triangle with $m (\angle C) = 90^\circ$ and the points $D \in [AC], E\in [BC]$. Inside the triangle we construct the semicircles $C_1, C_2, C_3, C_4$ of diameters $[AC], [BC], [CD], [CE]$ and let $\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4$. Show that points $K, L, M, N$ are concyclic.
3.85 2006 JBMO Shortlist G1 (9)
3.86 2006 JBMO Shortlist G3 problem 2
3.87 2006 JBMO Shortlist G2 (10)
3.88 2006 JBMO Shortlist G5 (12)
3.89 2007 JBMO Shortlist G2 problem 2
3.90 2008 Greece JBMO TST P1
3.91 2008 JBMO Shortlist G3 problem 2
3.93 2009 JBMO Shortlist G4 problem 1
3.17 1998 JBMO problem 2 (GRE)
Let ${ABCDE }$ be a convex pentagon such that ${AB = AE = CD = 1, \angle ABC = \angle DEA = 90^\circ }$ and ${BC + DE = 1 }$. Compute the area of the pentagon.
3.18 2001 JBMO problem 2 (BUL)
Let ${ABC }$ be a triangle with ${\angle C = 90^\circ }$ and ${CA \ne CB }$. Let ${CH }$ be an altitude and ${CL }$ be an interior angle bisector. Show that for ${X \ne C }$ on the line ${CL }$, we have ${\angle XAC \ne \angle XBC }$. Also show that for ${Y \ne C }$ on the line ${CH }$ we have ${\angle Y AC \ne \angle YBC }$.
3.19 2001 JBMO problem 3 (GRE)
Let ${ABC}$ be an equilateral triangle and ${D, E }$ points on the sides ${ [AB]}$ and ${ [AC] }$ respectively. If ${DF, EF }$ (with ${F \in AE, G \in AD }$) are the interior angle bisectors of the angles of the triangle ${ADE }$, prove that the sum of the areas of the triangles ${DEF }$ and ${DEG }$ is at most equal with the area of the triangle ${ABC}$. When does the equality hold?
3.20 2002 JBMO Shortlist G4 problem 1 (GRE)
The triangle ${ABC }$ has ${CA = CB }$. ${P }$ is a point on the circumcircle between ${A }$ and ${B }$ (and on the opposite side of the line ${AB }$ to ${C}$). ${D }$ is the foot of the perpendicular from ${C }$ to ${PB}$. Show that ${PA + PB = 2 \cdot PD }$.
3.21 2002 JBMO Shortlist G6 problem 2 (CYP)
Two circles with centers ${O_1}$ and ${O_2}$ meet at two points ${A}$ and ${B}$ such that the centers of the circles are on opposite sides of the line ${AB}$. The lines ${BO_1}$ and ${BO_2}$ meet their respective circles again at ${B_1}$ and ${B_2}$. Let ${M}$ be the midpoint of ${B_1B_2 }$. Let ${M_1, M_2}$ be points on the circles of centers ${O_1}$ and ${O_2}$ respectively, such that ${\angle AO_1M_1 =\angle AO_2M_2}$, and ${B_1}$ lies on the minor arc ${AM_1}$ while ${B }$ lies on the minor arc ${AM_2}$. Show that ${\angle MM_1B = \angle MM_2B}$.
3.22 - 3.413.42 2000 JBMO Shortlist 22
Consider a quadrilateral with ${\angle DAB = 60^\circ, \angle ABC = 90^\circ}$ and ${\angle BCD = 120^\circ }$. The diagonals ${AC}$ and ${BD}$ intersect at ${M}$. If ${MB = 1}$ and ${MD = 2}$, find the area of the quadrilateral ${ABCD}$.
3.43 2000 JBMO Shortlist 23
The point ${P}$ is inside of an equilateral triangle with side length ${10}$ so that the distance from ${P}$ to two of the sides are ${1}$ and ${3}$. Find the distance from ${P}$ to the third side.
3.44 2001 JBMO Shortlist 8 (FYROM)
Prove that no three points with integer coordinates can be the vertices of an equilateral triangle
3.45
3.46 2001 JBMO Shortlist 11 (ROM)
Consider a triangle ${ABC}$ with ${AB = AC}$, and ${D}$ the foot of the altitude from the vertex ${A}$. The point ${E}$ lies on the side ${AB}$ such that ${\angle ACE = \angle ECB = 18^\circ }$. If ${AD = 3}$, find the length of the segment ${CE}$.
Sorin Peligrad, Pitești
3.47 2002 JBMO Shortlist G1 (8)
Let ${ABC}$ be a triangle with centroid ${G}$ and ${A_1,B_1,C_1}$ midpoints of the sides ${BC,CA,AB}$. A parallel through ${A_1}$ to ${BB_1}$ intersects ${B_1C_1}$ at ${F}$. Prove that triangles ${ABC}$ and ${FA_1A}$ are similar if and only if quadrilateral ${AB_1GC_1}$ is cyclic.
3.48 2002 JBMO Shortlist G3 (10)
Let ${ABC}$ be a triangle with area ${S}$ and points ${D,E, F}$ on the sides ${BC,CA,AB}$. Perpendiculars at points ${D,E, F}$ to the ${BC,CA,AB}$ cut circumcircle of the triangle ${ABC at points (D_1,D_2), (E_1,E2), (F_1, F_2) }$. Prove that: ${|D_1B \cdot D_1C - D_2B \cdot D_2C| + |E_1A \cdot E_1C – E_2A \cdot E_2C| + |F_1B \cdot F_1A - F_2B \cdot F_2A| > 4S }$
3.49 2002 JBMO Shortlist G5 (11)
Let ${ABC}$ be an isosceles triangle with ${AB = AC}$ and ${\angle A = 20^\circ}$. On the side ${AC}$ consider point ${D}$ such that ${AD = BC}$. Find ${\angle BDC}$.
3.50 2002 JBMO Shortlist G7 (12)
Let ${ABCD}$ be a convex quadrilateral with ${AB = AD}$ and ${BC = CD}$. On the sides ${AB,BC,CD,DA}$ we consider points ${K,L,L_1,K_1}$ such that quadrilateral ${KLL_1K_1}$ is rectangle. Then consider rectangles ${MNPQ}$ inscribed in the triangle ${BLK}$, where ${M \in KB,N \in BL, P,Q \in LK}$ and ${M_1N_1P_1Q_1}$ inscribed in triangle ${DK_1L_1}$ where ${P_1 }$ and ${Q_1}$ are situated on the ${L_1K_1, M}$ on the ${DK_1}$ and ${N_1}$ on the ${DL_1}$. Let ${S, S_1, S_2, S_3}$ be the areas of the ${ABCD,KLL_1K_1,MNPQ,M_1N_1P_1Q_1}$ respectively. Find the maximum possible value of the expression: ${\frac{S_1+S_2+S_3}{S}}$
3.51 2002 JBMO Shortlist G8 (13)
Let ${A_1,A_2, ...,A_{2002}}$ be arbitrary points in the plane. Prove that for every circle of radius ${1}$ and for every rectangle inscribed in this circle, there exist ${3}$ vertices ${M,N, P}$ of the rectangle such that ${MA_1+MA_2+...+ MA_{2002}+NA_1+NA_2+...+NA_{2002}+PA_1+PA_2 +…+PA_{2002 }\ge 6006}$.
3.52 - 3.763.77 2005 JBMO Shortlist G1 (5)
Let $ABCD$ be an isosceles trapezoid with $AB=AD=BC, AB//CD, AB>CD$. Let $E= AC \cap BD$ and $N$ symmetric to $B$ wrt $AC$. Prove that the quadrilateral $ANDE$ is cyclic.
3.78 2005 JBMO Shortlist G2 problem 2
Let ${ABC}$ be an acute-angled triangle inscribed in a circle ${k}$. It is given that the tangent from ${A}$ to the circle meets the line ${BC}$ at point ${P}$. Let ${M}$ be the midpoint of the line segment ${AP}$ and ${R}$ be the second intersection point of the circle ${k}$ with the line ${BM}$. The line ${PR}$ meets again the circle ${k}$ at point ${S}$ different from ${R}$. Prove that the lines ${AP}$ and ${CS}$ are parallel.
3.79 2005 JBMO Shortlist G3 (6)
Let $ABCDEF$ be a regular hexagon and $M\in (DE)$, $N\in(CD)$ such that $m (\widehat {AMN}) = 90^\circ$ and $AN = CM \sqrt {2}$. Find the value of $\frac{DM}{ME}$.
3.80 2005 JBMO Shortlist G7 (10)
Let $ABCD$ be a parallelogram.
$P \in (CD), Q \in (AB), M= AP \cap DQ, N=BP \cap CQ, K=MN \cap AD, L= MN \cap BC$.
Prove that $BL=DK$.
3.81 2004 JBMO Shortlist G1 (7)
Two circles $C_1$ and $C_2$ intersect in points $A$ and $B$. A circle $C$ with center in $A$ intersect $C_1$ in $M$ and $P$ and $C_2$ in $N$ and $Q$ so that $N$ and $Q$ are located on different sides wrt $MP$, and $AB> AM$. Prove that $\angle MBQ = \angle NBP$.
3.82 2004 JBMO Shortlist G3 (9)
Let $ABC$ be a triangle inscribed in circle $C$. Circles $C_1, C_2, C_3$ are tangent internally with circle $C$ in $A_1, B_1, C_1$ and tangent to sides $[BC], [CA], [AB]$ in points $A_2, B_2, C_2$ respectively, so that $A, A_1$ are on one side of $BC$ and so on. Lines $A_1A_2, B_1B_2$ and $C_1C_2$ intersect the circle $C$ for second time at points $A’,B’$ and $C’$, respectively. If $ M = BB’ \cap CC’$, prove that $m (\angle MAA’) = 90^\circ$ .
3.83 2004 JBMO Shortlist G4 problem 2 Let ${ABC}$ be an isosceles triangle with ${AC = BC}$, let ${M}$ be the midpoint of its side ${AC}$, and let ${Z}$ be the line through ${C}$ perpendicular to ${AB}$. The circle through the points ${B, C}$, and ${M}$ intersects the line ${Z}$ at the points ${C}$ and ${Q}$. Find the radius of the circumcircle of the triangle ${ABC}$ in terms of ${m = CQ}$.
3.84 2004 JBMO Shortlist G5 (10)
Let $ABC$ be a triangle with $m (\angle C) = 90^\circ$ and the points $D \in [AC], E\in [BC]$. Inside the triangle we construct the semicircles $C_1, C_2, C_3, C_4$ of diameters $[AC], [BC], [CD], [CE]$ and let $\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4$. Show that points $K, L, M, N$ are concyclic.
3.85 2006 JBMO Shortlist G1 (9)
Let ${ABCD}$ be a trapezoid with ${AB // CD, AB > CD}$ and ${\angle A + \angle B = 90^\circ}$. Prove that the distance between the midpoints of the bases is equal to the semidifference of the bases.
3.86 2006 JBMO Shortlist G3 problem 2
Let ${ABCD}$ be an isosceles trapezoid inscribed in a circle ${C}$ with ${AB // CD, AB = 2CD}$. Let ${Q = AD \cap BC}$ and let ${P}$ be the intersection of the tangents to ${C}$ at ${B}$ and ${D}$. Calculate the area of the quadrilateral ${ABPQ}$ in terms of the area of the triangle ${PDQ}$
3.87 2006 JBMO Shortlist G2 (10)
The triangle ABC is isosceles with AB=AC, and ∠BAC<60∘. The points D and E are chosen on the side AC such that, EB=ED, and ∠ABD≡∠CBE. Denote by O the intersection point between the internal bisectors of the angles ∠BDC and ∠ACB. Compute ∠COD.
Let ${ABC}$ be an equilateral triangle of center ${O}$, and ${M \in BC}$. Let ${K,L}$ be projections of ${M}$ onto the sides ${AB}$ and ${AC}$ respectively. Prove that line ${OM}$ passes through the midpoint of the segment ${KL}$.
3.89 2007 JBMO Shortlist G2 problem 2
Let$ABCD$be a convex quadrilateral with $\angle DAC=\angle BDC={{36}^{o}},\angle CBD={{18}^{o}}$ and $\angle BAC={{72}^{o}}$. If $P$ is the point of intersection of the diagonals $AC$ and $BD$, find the measure of $\angle AP D$.
3.90 2008 Greece JBMO TST P1
Given a point $A$ that lies on circle $c(o,R)$ (with center $O$ and radius $R$). Let $(e)$ be the tangent of the circle $c$ at point $A$ and a line $(d)$ that passes through point $O$ and intersects $(e)$ at point $M$ and the circle at points $B,C$ (let $B$ lie between $O$ and $A$). If $AM = R\sqrt3$ , prove that
a) triangle $AMC$ is isosceles.
b) circumcenter of triangle $AMC$ lies on circle $c$ .
3.91 2008 JBMO Shortlist G3 problem 2
The vertices $A$ and $B$ of an equilateral $\vartriangle ABC$ lie on a circle $k$ of radius $1$, and the vertex $C$ is inside $k$. The point $D \ne B$ lies on $k$, $AD = AB$ and the line $DC$ intersects $k$ for the second time in point $E$. Find the length of the segment $CE$.
3.92 2003 JBMO Shortlist G4 problem 3 (BUL)
Let ${D, E, F}$ be the midpoints of the arcs ${BC, CA, AB}$ on the circumcircle of a triangle ${ABC}$ not containing the points ${A, B, C}$, respectively. Let the line ${DE}$ meets ${BC}$ and ${CA}$ at ${G}$ and ${H}$, and let ${M}$ be the midpoint of the segment ${GH}$. Let the line ${FD}$ meet ${BC}$ and ${AB}$ at ${K}$ and ${J}$, and let ${N}$ be the midpoint of the segment ${KJ}$.
a) Find the angles of triangle ${DMN}$,
b) Prove that if ${P}$ is the point of intersection of the lines ${AD}$ and ${EF}$, then the circumcenter of triangle ${DMN}$ lies on the circumcircle of triangle ${PMN}$.
Ch. Lozanov
Let ${ABCDE}$ be convex pentagon such that ${AB+CD=BC+DE}$ and ${k}$ half circle with center on side ${AE}$ that touches sides ${AB, BC, CD}$ and ${DE}$ of pentagon, respectively, at points ${P, Q, R}$ and ${S}$ (different from vertices of pentagon). Prove that $PS\parallel AE$.
Θα συνεχιστεί και με το Μαθηματικοί Διαγωνισμοί 2 μελλοντικά.
υπό κατασκευή:
3.1-3.42
3.433.44 1998 Mediterranean MO p3
In a triangle $ABC$, $I$ is the incenter and $D,E, F$ are the points of tangency of the incircle with $BC,CA,AB$, respectively. The bisector of angle $BIC$ meets $BC$ at $M$, and the line $AM$ intersects $EF$ at $P$. Prove that $DP$ bisects the angle $FDE$.
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3.62 2010 IMO p4
Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$). The lines $AP$, $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$, $L$, respectively $M$. The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$. Show that from $SC = SP$ follows $MK = ML$.
by Marcin E. Kuczma, Poland
3.63 2006 Mediterranean MO p2
Let $P$ be a point inside a triangle $ABC$, and $A_1B_2,B_1C_2,C_1A_2$ be segments passing through $P$ and parallel to $AB, BC, CA$ respectively, where points $A_1, A_2$ lie on $BC, B_1, B_2$ on $CA$, and $C_1,C_2$ on $AB$. Prove that $ \text{Area}(A_1A_2B_1B_2C_1C_2) \ge \frac{1}{2}\text{Area}(ABC)$
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3.75 1985 IMO p5
A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$
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3.86 1998 IMO Shortlist G2
Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.
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3.95 2005 IMO Shortlist G6
Let $ABC$ be a triangle, and $M$ the midpoint of its side $BC$. Let $\gamma$ be the incircle of triangle $ABC$. The median $AM$ of triangle $ABC$ intersects the incircle $\gamma$ at two points $K$ and $L$. Let the lines passing through $K$ and $L$, parallel to $BC$, intersect the incircle $\gamma$ again in two points $X$ and $Y$. Let the lines $AX$ and $AY$ intersect $BC$ again at the points $P$ and $Q$. Prove that $BP = CQ$
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3.98 2003 All- Russian MO 10 p7
In a triangle $ABC, O$ is the circumcenter and $I$ the incenter. The excircle $\omega_a$ touches rays $AB,AC$ and side $BC$ at $K,M,N$, respectively. Prove that if the midpoint $P$ of $KM$ lies on the circumcircle of $\triangle ABC$, then points $O,N, I$ lie on a line.
Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle.
M. Kungojin.
3.100 2012 IMO p5
Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$.
Show that $MK=ML$.
by Josef Tkadlec, Czech Republic
3.101 2011 IMO p6Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a, \ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a, \ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$.
by Japan
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