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MAN IS KID

Source: USAMO 2017 P3, Evan Chen

April 19, 2017

USA(J)MOUSAMO2017 USAMOgeometryHi

Problem Statement

Let

ABC

be a scalene triangle with circumcircle

\Omega

and incenter

I

. Ray

AI

meets

\overline{BC}

at

D

and meets

\Omega

again at

M

; the circle with diameter

\overline{DM}

cuts

\Omega

again at

K

. Lines

MK

and

BC

meet at

S

, and

L_2

is the midpoint of

\overline{IS}

. The circumcircles of

\overline{IL_1}

and

\triangle MAN

intersect at points

L_1

and

L_2

. Prove that

\Omega

passes through the midpoint of either

\overline{IL_1}

or

\overline{IL_2}

.
Proposed by Evan Chen

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