is the midpoint of the arc 
,
,
,
be the 
intersects 
.
is perpendicular to 
,
is the incenter of 
consisting of 
consecutive non-negative integers can be partitioned in two disjoint sets 
and 
each containing 
elements so that 
for 
of all sequences of integers 
such that 
and let 
.![\[\sum_{s\in A} M(s)=(n+1)^{k+1}-(1^k+2^k+\ldots +(n+1)^k).\]](http://latex.artofproblemsolving.com/7/6/e/76e7fce0da6bb9537b73d8699f7a21a6b8cb61fa.png)
with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each 
along a great circle of 
be a positive integer not divisible by 3. Prove that there are infinitely many positive integers 
such that 
and 
are divisible by 
is the sum of digits of 
.
be a prime number, and 
be a sequence defined by 
and 
,
.
terms of the sequence is not prime.
![\[\max(|\sin x|,|\sin (x+1)|)>\frac13.\]](http://latex.artofproblemsolving.com/e/d/e/ede9b524370c457fe41038cca5663e3c9bdaf320.png)
of lattice points with positive coordinates in the plane. For each point 
from ![\[S(P)=\{(a+b,c)\mid c\in\mathbb{Z}, \, c>a+b\}.\]](http://latex.artofproblemsolving.com/3/d/0/3d091c92ad9bf8644c59b77a1ad0209c41765093.png)
Show that there is no colouring of the points in 
in the plane of a square 
for which ![\[\max (PA, PC)=\frac1{\sqrt2}(PB+PD).\]](http://latex.artofproblemsolving.com/3/9/8/3984b1970837a85af0bac93fd711843630a1fd99.png)
,![\[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]](http://latex.artofproblemsolving.com/5/d/f/5df80f885e63852e4f53a6145dafd18d63dc3526.png)
is divisible by 
.
,
,
,
,
.
has two roots 
and 
.
and 
.
Y by DarintheBoy, goldendog, froyocat, Creeperboat, Squidget, HWenslawski, Sedro, Yrock, jkim0656, fastgoodboy, BookExplorer, Yiyj1, sangriabeaver5
Wow I can't believe I'm at 2000 posts already! I guess this also celebrates my (late) 3 year anniversary on AoPS!
um i guess i share my story (most problems are written by me)
1st grade -- i forgot ok
2nd grade -- i was at public school like the regular kids, there i started loving basketball and i was already working on prealgebra level stuff
3rd grade -- started aopsi skipped to calculus anyway so here i was still in public school and I started algebra a, but made no progress so my parents asked me if I want to homeschool (lucky they didn't force me), and i said yes
4th grade -- Finished intro to programming with python, got 13 on amc 8 (skull), finished all intro courses except intro to number theory. i was shaky on lots of concepts and i had to do a review (by myself with the aops books) sometimes.
5th grade -- finished intermediate programming with python but somehow failed usaco (the score shall be undisclosed), i started learning some basic C++, and finished all intro courses and im doing intermediate algebra and intermediate number theory now
. I got 21 on amc 8 (improvement but no dhr) and 72 on amc 10a (buh). oh and by the way I'm still playing basketball in my rec league 
for more example problems go to my two mock amc 8's and keep an eye on my other mock im about to create!
anyway thanks aops for 2000 posts it helped me learn so much
p.s. lots of information i didn't share but this is the general idea (also pls upvote! if i reach 10 upvotes i will create something special here)
um i guess i share my story (most problems are written by me)
1st grade -- i forgot ok
2nd grade -- i was at public school like the regular kids, there i started loving basketball and i was already working on prealgebra level stuff
3rd grade -- started aops
4th grade -- Finished intro to programming with python, got 13 on amc 8 (skull), finished all intro courses except intro to number theory. i was shaky on lots of concepts and i had to do a review (by myself with the aops books) sometimes.
5th grade -- finished intermediate programming with python but somehow failed usaco (the score shall be undisclosed), i started learning some basic C++, and finished all intro courses and im doing intermediate algebra and intermediate number theory now
. I got 21 on amc 8 (improvement but no dhr) and 72 on amc 10a (buh). oh and by the way I'm still playing basketball in my rec league for more example problems go to my two mock amc 8's and keep an eye on my other mock im about to create!
anyway thanks aops for 2000 posts it helped me learn so much
p.s. lots of information i didn't share but this is the general idea (also pls upvote! if i reach 10 upvotes i will create something special here)
This post has been edited 2 times. Last edited by evt917, Feb 26, 2025, 4:27 AM
It will be a long time till I get my 2000th post.
