[/quote]
,
,
,
be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle 
is tangent to sides 
and 
at 
and 
respectively, and 
is the circumcenter of 
.
.
denote the maximum number of 
-
board such that each 
such that![\[
f(N) \leq cN^2
\]](http://latex.artofproblemsolving.com/7/4/8/748109c3d63203f9a67a167434e10a909f38e83f.png)
for all positive integers 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
such that for all positive real numbers 
and 
,![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](http://latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
such that for all 
and 
,
such that 
or 
,![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](http://latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
be distinct real numbers such that![\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]](http://latex.artofproblemsolving.com/5/1/e/51eef77358db2b6c98f0fb0fb4e30db837a03642.png)
Find the maximum possible value of 
.
.The solution to 
is only 
for some 
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
be the orthocenter of acute triangle 
be the foot of the altitude from 
to 
be the reflection of 
.
intersects line 
and 
.
.
Y by AIME12345, JasperL, kingofgeedorah, 62861, Magikarp1, Generic_Username, budu, mathisawesome2169, checkmatetang, laegolas, speck, Mudkipswims42, rkm0959, skipiano, itised, zxcv1337, anantmudgal09, ThisIsASentence, algebra_star1234, PiDude314, champion999, Plasma_Vortex, artsolver, atmchallenge, Kaladesh, PcelicaMaja, CaptainFlint, AlgebraFC, brianapa, huricane, Problem_Penetrator, leequack, kk108, countdown1234, mathwizard888, soojoong, MinionSV, don2001, niyu, parmenides51, amar_04, AmirKhusrau, OlympusHero, myh2910, Adventure10
"If it is GGG, it will be GG..."
The contest is over!
Welcome, everyone! Over the last few weeks, I have had the pleasure of creating another olympiad geometry contest. My first one can be found here.
As users last time complained about the difficulty of 1/4, I've intentionally made 1/4 around JMO 1/4 level; 2/5/3/6 are closer to USAMO than USAJMO.
I'd like to thank users Generic_Username, AlgebraFC, MathSlayer4444, InCtrl, champion999, joey8189681, braniac1 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 have already recruited enough testsolvers, and I'd like to thank all who signed up to do so. Their usernames can be found below. Unfortunately, as JMO day I demonstrated, I still don't consider myself adequately good at bash to grade bash solutions, so I need people willing to grade ugly, ugly solutions. PM me if you want to grade bash solutions, or just post below!
See this post about time limits.
Congratulations to EulerMacaroni, tastymath75025, SAUDITYA, rkm0959, mofumofu, and huricane for winning the contest with perfect scores, and congratulations to PcelicaMaja and B.J.W.T for earning second place!
The contest is over!
Welcome, everyone! Over the last few weeks, I have had the pleasure of creating another olympiad geometry contest. My first one can be found here.
As users last time complained about the difficulty of 1/4, I've intentionally made 1/4 around JMO 1/4 level; 2/5/3/6 are closer to USAMO than USAJMO.
I'd like to thank users Generic_Username, AlgebraFC, MathSlayer4444, InCtrl, champion999, joey8189681, braniac1 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 have already recruited enough testsolvers, and I'd like to thank all who signed up to do so. Their usernames can be found below. Unfortunately, as JMO day I demonstrated, I still don't consider myself adequately good at bash to grade bash solutions, so I need people willing to grade ugly, ugly solutions. PM me if you want to grade bash solutions, or just post below!
See this post about time limits.
Congratulations to EulerMacaroni, tastymath75025, SAUDITYA, rkm0959, mofumofu, and huricane for winning the contest with perfect scores, and congratulations to PcelicaMaja and B.J.W.T for earning second place!
This post has been edited 36 times. Last edited by wu2481632, Jun 21, 2017, 4:09 AM
