Birthday power

by shiningsunnyday, Apr 15, 2017, 2:56 AM

2009 USAMO 1 wrote:

Given circles $\omega_1$ and $\omega_2$ intersecting at points $X$ and $Y$ , let $\ell_1$ be a line through the center of $\omega_1$ intersecting $\omega_2$ at points $P$ and $Q$ and let $\ell_2$ be a line through the center of $\omega_2$ intersecting $\omega_1$ at points $R$ and $S$ . Prove that if $P, Q, R$ and $S$ lie on a circle then the center of this circle lies on line $XY$ .

Solution

Tidbit

USAMO

geometry

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That's an example in EGMO btw

by MathAwesome123, Apr 15, 2017, 3:01 AM

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Wait actually that solution is incorrect.

RIP. Good point - will fix later

This post has been edited 1 time. Last edited by shiningsunnyday, Apr 15, 2017, 3:17 AM

by MathAwesome123, Apr 15, 2017, 3:03 AM

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Even so, that additional complication was the hard part of the problem anyway.

by MathAwesome123, Apr 15, 2017, 3:08 AM

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The PoP solution avoid configuration issues.

by First, Apr 15, 2017, 3:14 AM

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this guy is an absolute legend. much love wherever you are Michael
by LeonidasTheConquerer, Aug 5, 2024, 9:37 PM
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can i be a contributor please?
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by samrocksnature, Jul 16, 2021, 8:25 PM
2021 post
by the_mathmagician, May 5, 2021, 3:28 PM
Let $ABC$ be an equilateral triangle of side length $1$ . Let $D$ be the point such that $C$ is the midpoint of $BD$ , and let $I$ be the incenter of triangle $ACD$ . Let $E$ be the point on line $AB$ such that $DE$ and $BI$ are perpendicular. $ \
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Nice blog!
by User526797, Jan 12, 2020, 4:48 PM
oh my gosh it's been so longggggg.... contrib? what does that mean?
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2019 post
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contrib /charmander
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all contrib requests added I usually get to posting every night after running on the treadmill and showering, posting my favorite problem(s) I did during the day.
by shiningsunnyday, Jun 16, 2016, 4:37 AM
Since problems is plural, he might post multiple problems through different posts in one day...
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hi can i have contrib?
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Hello it is called problems of the DAY
by Temp456, Jun 15, 2016, 8:17 PM
10th shout lol
How often will this blog be updated?
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why are we counting shouts

contrib pls
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eighth shout
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Can I be contrib?
Though probably not because it's not like I'll post anything anyways :/ :/
by zephyrcrush78, Jun 15, 2016, 1:20 AM
sixth shout

new goal: get the shouts that are congruent to 1 mod 5
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fifth shout >
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270 shouts

Problems of the day

Birthday power

by shiningsunnyday, Apr 15, 2017, 2:56 AM

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