[/quote]
,
be a scalene triangle with circumcircle 
and incenter 
.
meets 
at 
and meets 
;
cuts 
.
and 
meet at 
,
is the midpoint of 
.
and 
intersect at points 
and 
.
or 
.
be an integer. Find all positive real solutions to the following system of 
equations:
,
,
,![\[ \frac {1}{a^3 + b^3 + abc} + \frac {1}{b^3 + c^3 + abc} + \frac {1}{c^3 + a^3 + abc} \leq \frac {1}{abc}
\]](http://latex.artofproblemsolving.com/f/a/9/fa964e29de91f0e2f3cb32e335a52d4521bc38fd.png)
is rotated 
counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has 
-
,
,
,
are positive integers, and 
.
prove that
be the circumcenter and 
the orthocenter of an acute triangle 
Prove that the area of one of the triangles 
and 
is equal to the sum of the areas of the other two.
with any line through the centroid 
and I have no clue as to why that is true. Plz help
such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
board, there is initially a tree of height 
.
.
.
be its 
-
be midpoint of arc 
,
be the intersection of the line 
with 
,
be the intersection of the angle bisector of 
with the line 
,the intersection of line 
with 
and 
,the intersection of 
and 
such that 
.
are collinear. Who has a winning strategy?
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
.
meets 
and intersects 
,
.
or 
is tangent to 
,
be a point on the segment 
with the diagonal 
.
at point 
.
such that 
.
and the segment 
.
and 
are concyclic.
that satisfy the following condition: for all 
we have ![\[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]](http://latex.artofproblemsolving.com/e/8/5/e85c5b249ac8e75b45a525b8786414798a4921fe.png)
![\[ \begin{tabular}{l|cccccccc}
Username & P1 & P2 & P3 & P4 & P5 & P6 & Total \\ \hline
RC. & 7 & 7 & 7 & 7 & 7& 7 & 42 \\
Hydrogen-Helium & 7 & 7 & 7 & 7 & 7 & 7 & 42 \\
\end{tabular}\]](http://latex.artofproblemsolving.com/b/1/8/b1823ff76aa93cb1173712268a7111ebbffc585c.png)
Y by Generic_Username, KateL, anantmudgal09, checkmatetang, PiDude314, niyu, brainiac1, skipiano, yegkatie, WireCat, Kagebaka, soojoong, Vfire, atmchallenge, AIME12345, SAUDITYA, Assmit, tenplusten, Mudkipswims42, FedeX333X, RamanConjecture, mira74, GeneralCobra19, oceanair, SAM8, qwerty733, roseylily, laegolas, thedoge, Snakes, franchester, DrMath, Plasma_Vortex, 62861, pifinity, jangkun217, bluephoenix, enhanced, N.G_using_P.G, speck, mukes070304, bigmath, targo___, Aryan-23, amar_04, AmirKhusrau, OlympusHero, Toinfinity, myh2910, parmenides51, centslordm, Adventure10
"If it is GGG, it will be GG..."
Hi everyone! Over the last few weeks, I have had the pleasure of creating another olympiad geometry contest. My first one can be found here and my second one can be found here.
I believe the contest is somewhere between the USAJMO and USAMO in difficulty.
Sign up for the contest by posting in the thread, please. The contest will be released in approximately a week, after testsolvers have finished looking over it.
I'd like to thank users Generic_Username, AlgebraFC, MathSlayer4444, InCtrl, champion999, joey8189681, brainiac1 for being there to support me in the creation of this contest. As well as complaining about the difficulty of drawing the diagrams, they also provided moral support and, in some cases, testsolved problems.
I only currently have three testsolvers, so I would like at least one more to guarantee quality; message me or post in-thread if you would like to testsolve. I have enough testsolvers now, yay! I'm also really lazy and still haven't properly learned enough bash (I only know the basics of complex and bary) so signups for grading bashes would be really, really appreciated.
Test deadline is December 1.
See this post about time limits.
Hi everyone! Over the last few weeks, I have had the pleasure of creating another olympiad geometry contest. My first one can be found here and my second one can be found here.
I believe the contest is somewhere between the USAJMO and USAMO in difficulty.
Sign up for the contest by posting in the thread, please. The contest will be released in approximately a week, after testsolvers have finished looking over it.
I'd like to thank users Generic_Username, AlgebraFC, MathSlayer4444, InCtrl, champion999, joey8189681, brainiac1 for being there to support me in the creation of this contest. As well as complaining about the difficulty of drawing the diagrams, they also provided moral support and, in some cases, testsolved problems.
Test deadline is December 1.
See this post about time limits.
This post has been edited 19 times. Last edited by wu2481632, Dec 2, 2017, 3:11 PM
.
)
