[/quote]
,
different icons.
be a positive integer and let 
be a strictly increasing sequence of 
positive real numbers with sum equal to 2. Let 
be a subset of 
such that the value of![\[
\left|1-\sum_{i \in X} a_{i}\right|
\]](http://latex.artofproblemsolving.com/2/5/1/2515e074de1782f38453e1fce8ab7c2ec7cfe4c0.png)
is minimised. Prove that there exists a strictly increasing sequence of 
with sum equal to 2 such that![\[
\sum_{i \in X} b_{i}=1.
\]](http://latex.artofproblemsolving.com/3/7/4/37410820fc4c546f11d57212cfce53a7ec4941e3.png)
be a scalene triangle with circumcircle 
,
at 
.
meets 
and 
.
intersects the 
-
,
,
.
passes through either 
be positive real numbers such that : 
. Prove that : 
![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
be relatively prime nonconstant polynomials. Show that there can be at most three real numbers 
such that 
is the square of a polynomial.
be real numbers less than 2.Then prove that 
and 
intersect each other at 
and 
.
and 
respectively. Let 
and 
be the reflection of 
intersect circles 
and 
(both distinct from 
is the second tangent to 
.
with center 
,
are all different and lie on line 
be the points of intersection of 
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
and 
are different and intersect at the point 
.
be the diameter of semicircle 
.
and is tangent to 
bisects 
,
,
,
,
.
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.
