[/quote]
,
:o
so 
and since 
is just 
therefore 
so 
and 
. Prove that
. Prove that 
and 
. Prove that
Where 
![$ a,b\in [0,1] $](http://latex.artofproblemsolving.com/9/7/5/975edfc0cf5316d78283b313c6b5cfbdd7b92b6e.png)
. Prove that
Where 
Where 
where 
are lengths of parallel sides and 
are other two sides.
be real numbers satisfying 
and ![\[a_{i+1} - 2a_i + a_{i-1} = a^2_i\]](http://latex.artofproblemsolving.com/c/d/8/cd878e4ac230853182da75c54dcbf7be05b305bc.png)
for 
Prove that 
for 
,
letters 
can be placed in each cell of a table with 
columns, and has the following property: For any two distinct rows in the table, there exists at most one column, such that the entries of these two rows in such a column are the same letter.
be an continuous function defined in ![$\text{[0,2015]},f(0)=f(2015)$](http://latex.artofproblemsolving.com/c/9/9/c99f63e726b504aea8c1a65ba2926155a873effa.png)
pairs of 
such that 
Y by LuminousWolverine, eggymath, tofubear, iralavu, PikaPika999, Yolandayu, engineer48
Hello AOPS,
As we enter the New Year, the most exciting part is figuring out the mathematical connections to the number we have now temporally entered
Here are some facts about 2025:
If anyone has any more facts about 2025, enlighted the world with a new appreciation for the year
(I got some of the facts from this video)
As we enter the New Year, the most exciting part is figuring out the mathematical connections to the number we have now temporally entered
Here are some facts about 2025:
If anyone has any more facts about 2025, enlighted the world with a new appreciation for the year
(I got some of the facts from this video)
This post has been edited 2 times. Last edited by Existing_Human1, Jan 1, 2025, 8:39 PM
since it looks like 
I wanna hit 1 million like Mango247 but just without a bot. If you look at the threads that I post on, most of the posts will have a thumbsup by me 
POSTS 
!
