IMO 2013 #4

by Wolstenholme, Oct 27, 2014, 8:52 PM

Let $ABC$ be an acute triangle with orthocenter $H$ , and let $W$ be a point on the side $BC$ , lying strictly between $B$ and $C$ . The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$ , respectively. Denote by $\omega_1$ is the circumcircle of $BWN$ , and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$ . Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$ , and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$ . Prove that $X,Y$ and $H$ are collinear.

Proof:

Let $P \ne W$ be the second intersection of the two circles. Since quadrilateral $BCMN$ is cyclic (with diameter $BC$ ) we have that the radical center of it and the two circles constructed in the problem is $A$ . Therefore $A \in WP$ . Now since $\angle{WPX} = \angle{XPY} = 90$ we have that $X, P, Y$ are collinear. Therefore $P$ is the projection of $A$ onto $XY$ . Hence, it suffices to show that $\angle{APH} = 90$ , which is equivalent to showing that $P$ lies on the circumcircle of quadrilateral $AMHN$ . But this follows immediately from Miquel's Theorem, so we are done.

This post has been edited 1 time. Last edited by Wolstenholme, Oct 27, 2014, 9:09 PM

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It should be 2013 not 2014

Also, how do you get so good? It seems as if you can solve any IMO problem.

by TheMaskedMagician, Oct 27, 2014, 9:01 PM

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Well, I became OK after I just practiced a lot. I read tons of books and articles, and just did a lot of problems. But, I'm probably not as good as you think I am...

I'm doing these problems with no time limit, from the comfort of my own home/school, and without the pressure of the actual competition. Also, I definitely can't solve all of them - I didn't solve IMO 2014 #6, and in fact, I'm probably going to skip IMO 2013 #6 as well

If I were to actually go to the IMO right now (and obviously this is based a lot on luck) I think I would get a bronze or a silver (maybe I'm just underestimating my self, who knows?

)

This post has been edited 1 time. Last edited by Wolstenholme, Oct 27, 2014, 9:13 PM

by Wolstenholme, Oct 27, 2014, 9:13 PM

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glad to have found a fellow chipotle lover <3
by nukelauncher, Aug 13, 2020, 6:40 AM
the random chinese tst problem is the only thing I read, but I'll assume your blog is nice and give you a shout even though you probably never use aops anymoer
by fukano_2, Jun 14, 2020, 6:24 AM
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by AopsUser101, Jan 29, 2020, 8:27 PM
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Hi. Nice Blog!
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by songssari, Jun 12, 2016, 8:21 AM
shouts make blogs happier
by briantix, Mar 18, 2016, 9:57 PM
You were just featured on AoPS's facebook page.
by mishka1980, Sep 12, 2015, 10:33 PM
This is late, but where is the ARML results post?
by donot, Aug 31, 2015, 11:07 PM
"I am Sam"
"Sam I am"
by mathwizard888, Aug 12, 2015, 9:13 PM
HW $\textcolor{white}{}$
by Eugenis, Apr 20, 2015, 10:10 PM
Uh-oh ARML practice is Thursday... I should start the homework.
by nosaj, Apr 20, 2015, 12:34 AM
Yes I am Sam, and Chebyshev polynomials aren't trivial, although they do make some problems trivial
by Wolstenholme, Apr 15, 2015, 10:00 PM
How are Chebyshev Polynomials trivial?
by nosaj, Apr 13, 2015, 4:10 AM
Are you Sam?
by Eugenis, Apr 4, 2015, 2:05 AM
@Brian: yes, yes I did #whoneedsalgskillz?

@gauss1181; hey!
by Wolstenholme, Mar 1, 2015, 11:25 PM
hello!!!
by gauss1181, Nov 27, 2014, 12:19 AM
Hi Wolstenholme did you actually use calc on that tstst problem
by briantix, Aug 2, 2014, 12:25 AM

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IMO 2013 #4

by Wolstenholme, Oct 27, 2014, 8:52 PM

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