[/quote]
,
is not equal to one. I looked it up, and it said 
.
be a scalene triangle, and let 
be a point on side 
satisfying 
.
and 
are points inside 
and 
are similar and quadrilaterals 
and 
are cyclic. Let lines 
and 
meet at 
and lines 
and 
meet at 
.
and 
are parallel.
which is a perfect square is of the form 
for 
be an integer. Prove that there exists a positive integer 
with the following property: For each positive integer 
,
(possibly depending on 
)
is divisible by 
.
and 
be positive divisors of 
.
is not divisible by 
.
,
.
with integer coefficients such that for any positive integer 
,
is positive and 
is called 
if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
.
of positive integers with ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
the set of all positive integers. Find all functions 
such that for all positive integers 
and 
is nonzero and divides 
.
,
.
.
lie on segments 
respectively. Points 
are such that 
and 
are equlateral, 
and 
lie on opposite sides of 
,
and 
lie on the opposite siedes of 
.
,
,
,
,
.
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.
