[/quote]
,
such that![\[f(f(f(\frac{x+y}{2}))+x+y)=f(x)+f(y)+f(\frac{x+y}{2})\]](http://latex.artofproblemsolving.com/6/2/4/624afa78b68f810aaaecf0e1e2251047beeddc36.png)
for all rational numbers 
.
in term of a function 
such that 
such that![\[2^{n!}+1\mid 2^{m!}+19\]](http://latex.artofproblemsolving.com/8/7/e/87ede5616cffcb33aa187f5d8a13827470f741b4.png)
,
Prove that
,
inequalities, these given comparisons satisfy the following three constraints:
and 
,
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
that satisfies all these conditions?
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
pairwise distinct positive integer numbers 
,![\[
\frac{a_i^2 + a_j^2}{a_i + a_j}
\]](http://latex.artofproblemsolving.com/3/5/3/3537ddec6d793d3831ba82a1e13b98d43b1c29bc.png)
students in the IMO team selection camp.
team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these 
be the circumcenter of an acute-angled triangle 
with 
.
meets the side 
at 
.
and 
are 
and 
,
and 
beyond 
,
and 
such that 
and 
.
is a rectangle if and only if 
.
be a convex pentagon such that 
.
is the circumcenter of triangle 
.
.
be the Fermat point of a 
.
,
,
are concurrent and the point of concurrence is 
.
at points 
,
,
be the midpoint of 
.
and 
intersect the circumcircle of triangle 
,
coincides with 
Y by bjump, EpicBird08, Awaycappybara, HamstPan38825, dolphinday, ihatemath123, Lionking212, Plasma_Vortex, megarnie, think4l, alan9535, r00tsOfUnity, peace09, Rounak_iitr, duo_duo, OronSH, resources, WalterMitchell, gladIasked, RetroTurtle, RaymondZhu, sargamsujit, YaoAOPS, Sedro, Leo.Euler, sky2025, pikapika007, awesomehuman, lethan3, Anchovy, llbaobao, Geometry285, AI216, anantmudgal09, AtharvNaphade, Math4Life7, MathLuis, UnknownMonkey, ryanbear, Amkan2022, peelybonehead, NTfish, juicetin.kim, Danielzh, Awesomeness_in_a_bun, justJen, zhoujef000, Significant, Turtwig113, Mogmog8, brianzjk, aops4ever1, songyanxin, arjken, mathmax12, ex-center, ChromeRaptor777, parmenides51, spiritshine1234, aidan0626, suvamkonar, chessboy_123, scannose, Alcumusgrinder07, cursed_tangent1434, NH14, NamelyOrange, hh99754539, roribaki, abeot, ESAOPS, YBSuburbanTea, pineapply, tigeryong, Puddles_Penguin, crazyeyemoody907, vrondoS, joemama_69_420, ultimatehero, fuzimiao2013, v4913, rg_ryse, Taco12, Silverblossom, Jndd, mal7896, AceCard, michaelwenquan, Bradygho, mathboy100, Ritwin, blueprimes, Aryan-23, zhenghua, NicoN9
Satisfactory. Keep cooking.
I'm happy to announce that after I think a 10-year hiatus or something, I've finally gotten back into the business of writing random mock exams like I did when I was a kid. Actually, just kidding, I didn't make any of the problems --- I just did a lot of editing.
The OTIS Mock AIME 2024 will be a 15-problem mock AIME showcasing some really nice problems set by current and past OTIS students. (It was actually pretty hard for me to choose the lineup because there weren't enough slots to use all the problems I liked; maybe next year.)
The details are written out at https://web.evanchen.cc/mockaime.html, but to highlight important info:
I'm happy to announce that after I think a 10-year hiatus or something, I've finally gotten back into the business of writing random mock exams like I did when I was a kid. Actually, just kidding, I didn't make any of the problems --- I just did a lot of editing.
The OTIS Mock AIME 2024 will be a 15-problem mock AIME showcasing some really nice problems set by current and past OTIS students. (It was actually pretty hard for me to choose the lineup because there weren't enough slots to use all the problems I liked; maybe next year.)
The details are written out at https://web.evanchen.cc/mockaime.html, but to highlight important info:
- The problems can be downloaded from the link above, or directly from https://web.evanchen.cc/exams/OTIS-Mock-AIME-2024.pdf.
- If you'd like to submit for scoring, you should do so by January 15 at 23:59. Please hold off on public spoilers before then.
- Solutions, statistics, and maybe some high scores will be published shortly after that.
This post has been edited 3 times. Last edited by v_Enhance, Dec 22, 2023, 8:13 PM
Zaahir Ali (15 points)
thanosaops (14 points)
tenth (13 points)
is a divisor of both 
and 
.
