[/quote]
,
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
,
lies on diagonal 
such that 
.
.
has 
,
,
.
.
be an integer. You have two 
boards. Each board contains the numbers 
to 
inclusive, one number per square, arbitrarily arranged on each board. A move consists of exchanging two rows or two columns on the first board (no moves can be made on the second board). Show that it is possible to make a sequence of moves such that for all 
and 
,
row and 
column of the first board is different from the number that is in the 
be a convex pentagon such that![\[ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.
\]](http://latex.artofproblemsolving.com/d/3/d/d3d9a82f4318190298a8f008d417a364e03f1fca.png)
The diagonals 
and 
meet at 
.
bisects the side 
.
,
,
is the incenter, 
-
.
lies on the external angle bisector of 
such that 
and 
.
lies on 
.
is tangent to 
and 
at 
,
is tangent to 
and both circles pass through the inside of triangle 
is the Midpoint od the arc 
be a sequence of positive real numbers, and 
be a positive integer, such that![\[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\]](http://latex.artofproblemsolving.com/f/1/2/f12c9c5e0f154549118a33a2f359329333f432a7.png)
and 
,![\[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]](http://latex.artofproblemsolving.com/b/0/b/b0b2bd4507621302eeee27ae50ed9c0afd72c6ef.png)
meet the circle again at 
parallel to 
.
is parallel to 
.
,
be positive real numbers, such that 
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
![\[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\]](http://latex.artofproblemsolving.com/1/e/8/1e8b3e2d9a4d0917ec622449b058702700542f22.png)
for positive integers 
,
be prime, let 
be a positive integer, show that ![\[ \gcd\left({p - 1 \choose n - 1}, {p + 1 \choose n}, {p \choose n + 1}\right) = \gcd\left({p \choose n - 1}, {p - 1 \choose n}, {p + 1 \choose n + 1}\right). \]](http://latex.artofproblemsolving.com/9/d/4/9d409ce186320241eb358b5fbbb8a5a66120899c.png)
,
,
,
,
.
has two roots 
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.
