[/quote]
,
be an integer. Ion draws a regular 
-
such that 
coprime, 
and 
.
be an acute-angled triangle and 
,
be the feet of the altitudes from 
,
,
,
,
,
respectively, lying outside of ![\[\frac{A_1A_2}{AA_1} = \frac{B_1B_2}{BB_1} = \frac{C_1C_2}{CC_1}.\]](http://latex.artofproblemsolving.com/d/f/a/dfa94e10638971d53066c20c4589a9c125059911.png)
If the intersections of 
and 
,
and 
,
and 
are 
,
respectively, prove that 
,
have a common point.
such that
,
,
,
denote the greatest common divisor of 
.
is an integer.
satisfying the equation![\[ 4x^2 - 3y^2 = 1 \]](http://latex.artofproblemsolving.com/d/2/8/d284c2c3a7d7fbe12a7988a4534df16e7adc8947.png)
such that 
.
and 
are standing in front of the room in which there is 
such that for any real value of 
we have:
Y by OronSH, Kempu33334, Sedro, bjump, megahertz13, scannose, dolphinday, megarnie, vincentwant, zhenghua, Quidditch, jocaleby1, carcool0, RuinGuard, happypi31415, blueprimes, Mogmog8, justJen, awesomeguy856, Ilikeminecraft, Danielzh, EpicBird08, Mathandski, ld414think, evt917, bachkieu, Awesomeness_in_a_bun, puffypundo, YaoAOPS, Sleepy_Head, KevinYang2.71, akliu, aidan0626, beishexi, elasticwealth, thinkcow, Shadow6885, deduck, mathleticguyyy, ravengsd, math_comb01, Alex-131, lpieleanu, NamelyOrange, clarkculus, centslordm, ehuseyinyigit, mathwiz_1207, khina, DhruvJha, gladIasked, Taco12, BR1F1SZ, tricky.math.spider.gold.1, Exponent11, CT17, LikMCAMC, Awesome_guy, alan9535, Tem8, Quentissential, zpwu007, NicoN9, bluelinfish, GeoKing, SnowPanda, Everestbaker, trk08, QueenArwen, CyclicISLscelesTrapezoid, Andyluo, anduran, samrocksnature, Arcticturn, cursed_tangent1434, Jndd, mathfan2020, qwerty123456asdfgzxcvb, Awesome_Twin1, vrondoS, Jack_w, golue3120, syndrome2027, Radio2, mariairam, cosinesine
Congratulations to all of you!!!!!!!
Alexander Wang
Hannah Fox
Karn Chutinan
Andrew Lin
Calvin Wang
Tiger Zhang
Good luck in Australia!
Alexander Wang
Hannah Fox
Karn Chutinan
Andrew Lin
Calvin Wang
Tiger Zhang
Good luck in Australia!
Have fun!
