[/quote]
,
and a positive integer 
is given.
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](http://latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](http://latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.
be a positive integer. For every positive integer 
the sequence 
is defined, where 
are integers. Among these 
with 
.
and 
are on 
and 
respectively such that 
.
and 
meet at 
.
is the incenter of 
and 
is the orthocenter of 
.
and 
are circles with diameter 
and 
,
and 
.
.
with 
with all the unit squares are initially white coloured. Every move, we can turn the color (from white to black or otherwise) from the 5 unit squares that form this T-pentomino which can be rotated or reflexed (see the image below). Determine all natural numbers 
such that all unit squares on the board can be made into all black after a finite number of moves.
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 
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 
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 
.
