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2017 USAMO Problems/Problem 3: Difference between revisions

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==Solution==
==Solution==
[[File:2017 USAMO 3.png|400px|right]]
[[File:2017 USAMO 3.png|400px|right]]
[[File:2017 USAMO 3a.png|400px|right]]
Let <math>X</math> be the point on circle <math>\Omega</math> opposite <math>M \implies  \angle MAX = 90^\circ, BC \perp XM.</math>
Let <math>X</math> be the point on circle <math>\Omega</math> opposite <math>M \implies  \angle MAX = 90^\circ, BC \perp XM.</math>


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<math>\angle XI'I = \angle XAI =  90^\circ \implies</math>  the points <math>X, I' ,</math> and <math>F</math> are collinear.
<math>\angle XI'I = \angle XAI =  90^\circ \implies</math>  the points <math>X, I' ,</math> and <math>F</math> are collinear.


<math>I'IDD'</math> is cyclic <math>\implies \angle I'D'M = \angle I'D'C + 90^\circ =  \angle I'ID + 90^\circ, \angle XFM = \angle I'FI = 90^\circ – \angle I'IF = 90^\circ – \angle I'ID  \implies</math>
<math>I'IDD'</math> is cyclic <math>\implies \angle I'D'M = \angle I'D'C + 90^\circ =  \angle I'ID + 90^\circ,</math>
<math>\angle XFM = \angle I'FI = 90^\circ – \angle I'IF = 90^\circ – \angle I'ID  \implies</math>


<math>\angle XFM +  \angle I'D'M = 180^\circ \implies I'D'MF</math> is cyclic.  
<math>\angle XFM +  \angle I'D'M = 180^\circ \implies I'D'MF</math> is cyclic.
Therefore point <math>F</math> lies on  <math>I_\omega (IDK).</math>
Therefore point <math>F</math> lies on  <math>I_\omega (IDK).</math>


In <math>\triangle FSX</math> <math>FA \perp SX, SI' \perp FX \implies I</math> is orthocenter of <math>\triangle FSX.</math>
<math>FA \perp SX, SI' \perp FX \implies I</math> is orthocenter of <math>\triangle FSX.</math>


<math>N</math> is midpoint <math>SI, M</math> is midpoint <math>SI, I</math> is orthocenter of <math>\triangle FSX, A</math> is root of height <math>FA \implies</math> circle <math>AMN</math> is the nine-point circle of <math>\triangle FSX \implies I'</math> lies on circle <math>AMN.</math>  
<math>N</math> is midpoint <math>SI, M</math> is midpoint <math>FI, I</math> is orthocenter of <math>\triangle FSX, A</math> is root of height <math>FA \implies AMN</math> is the nine-point circle of <math>\triangle FSX \implies I' \in AMN.</math>  


Let <math>N' = I_\omega (N) \implies R^2 = SN \cdot SN' = SI \cdot  SI' \implies \frac {SN'}{SI'} = \frac {SI}{SN} =2 \implies</math>
Let <math>N' = I_\omega (N) \implies R^2 = SN \cdot SN' = SI \cdot  SI' \implies</math>
<cmath>\frac {SN'}{SI'} = \frac {SI}{SN} =2 \implies</cmath>
<math>\angle XN'I' = \angle XSI' = 90^\circ – \angle AXI' = \angle IFX \implies N'XIF</math> is cyclic.  
<math>\angle XN'I' = \angle XSI' = 90^\circ – \angle AXI' = \angle IFX \implies N'XIF</math> is cyclic.  


Revision as of 04:38, 21 September 2022

Problem

Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I.$ Ray $AI$ meets $BC$ at $D$ and $\Omega$ again at $M;$ the circle with diameter $DM$ cuts $\Omega$ again at $K.$ Lines $MK$ and $BC$ meet at $S,$ and $N$ is the midpoint of $IS.$ The circumcircles of $\triangle KID$ and $\triangle MAN$ intersect at points $L_1$ and $L.$ Prove that $\Omega$ passes through the midpoint of either $IL_1$ or $IL.$

Solution

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Let $X$ be the point on circle $\Omega$ opposite $M \implies \angle MAX = 90^\circ, BC \perp XM.$

$\angle XKM = \angle DKM = 90^\circ \implies$ the points $X, D,$ and $K$ are collinear.

Let $D' = BC \cap XM \implies DD' \perp XM \implies$

$S$ is the orthocenter of $\triangle DMX \implies$ the points $X, A,$ and $S$ are collinear.

Let $\omega$ be the circle centered at $S$ with radius $R = \sqrt {SK \cdot SM}.$

We denote $I_\omega$ inversion with respect to $\omega.$

Note that the circle $\Omega$ has diameter $MX$ and contain points $A, B, C,$ and $K.$

$I_\omega (K) = M \implies$ circle $\Omega \perp \omega \implies C = I_\omega (B), X = I_\omega (A).$

$I_\omega (K) = M \implies$ circle $KMD \perp \omega \implies D' = I_\omega (D) \in KMD \implies$ $\angle DD'M = 90^\circ \implies$ the points $X, D',$ and $M$ are collinear.

Let $F \in AM, MF = MI.$ It is well known that $MB = MI = MC \implies$

$\Theta = BICF$ is circle centered at $M.$ $C = I_\omega (B) \implies \Theta \perp \omega.$

Let $I' = I_\omega (I ) \implies I' \in \Theta \implies \angle II'M = 90^\circ.$ $I' = I_\omega (I ), X = I_\omega (A ) \implies AII'X$ is cyclic.

$\angle XI'I = \angle XAI = 90^\circ \implies$ the points $X, I' ,$ and $F$ are collinear.

$I'IDD'$ is cyclic $\implies \angle I'D'M = \angle I'D'C + 90^\circ = \angle I'ID + 90^\circ,$ $\angle XFM = \angle I'FI = 90^\circ – \angle I'IF = 90^\circ – \angle I'ID \implies$

$\angle XFM + \angle I'D'M = 180^\circ \implies I'D'MF$ is cyclic.

Therefore point $F$ lies on $I_\omega (IDK).$

$FA \perp SX, SI' \perp FX \implies I$ is orthocenter of $\triangle FSX.$

$N$ is midpoint $SI, M$ is midpoint $FI, I$ is orthocenter of $\triangle FSX, A$ is root of height $FA \implies AMN$ is the nine-point circle of $\triangle FSX \implies I' \in AMN.$

Let $N' = I_\omega (N) \implies R^2 = SN \cdot SN' = SI \cdot SI' \implies$ \[\frac {SN'}{SI'} = \frac {SI}{SN} =2 \implies\] $\angle XN'I' = \angle XSI' = 90^\circ – \angle AXI' = \angle IFX \implies N'XIF$ is cyclic.

$I_\omega (NAI'M) = N'XIKF \implies I_\omega(F) = L \implies$ the points $F, L,$ and $S$ are collinear.

Point $I$ is orthocenter $\triangle FSX \implies XI \perp SF, \angle ILS = \angle SI'F = 90^\circ \implies$ the points $X, I, E,$ and $L$ are collinear.

$SAI'F = I_\omega (XIF) \implies AI \cdot IF = SI \cdot II' = AI \cdot \frac {IF}{2} = \frac {SI}{2} \cdot II' = AI \cdot IM = EI \cdot IX \implies AEMX$ is cyclic.

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