AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
and 
. Prove that
Let 
be reals such that 
. Prove that
real numbers 
, we can select a real number 
and transform the sequence into 
. This transformation can be performed multiple times, with each chosen real number potentially being different
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the grid.
cars from the grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid. that guarantees that Kritesh's job is possible?
be four positive integers in arithmetic progression (that is, 
) and with 
. Can the product 
ever be a number of the form 
for some 
and some 
, with 
?
, find all possible positive integer values of![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](http://latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
with 
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let 
be a point on such that 
Let 
be the intersection of 
and Prove that 
for which there exists an integer 
such that 
is a divisor of 
.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and .
are in arithmetic progression and 
then show that the product 
is not a perfect square
and 
be two sequences of positive numbers. Show that the following statements are equivalent:
of positive numbers such that 
and 
both converge;[/*]
converges.[/*], let 
be the number of positive divisors of . Evaluate
dividing by x
has 
different RREFs (Check the uploaded file) and the maximum integer among them is 
Find 
.
matrices which has only 
and 