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.
I can prove that the sequence 
is convergent by monotone bounded convergence theorem., or give the asymptotic representation of ? Any help and hints will welcome!
, such that 
Prove that :
equality cases : ?
be a positive integer, and consider the equation:
Answer the following questions. You may assume that 
is known.
Determine the number of real solutions of equation 
for each .
For 
, let 
be the segond largest real solution of .
Find 
such that 
Find 
, where 
is defined as
, such that 
and 
.
such that 
is prime, 
with 
and ?
be a positive integer. Egor has 
cards with the number “
” written on them, and cards with the number “
” written on them. Egor wants to paint each card red or blue so that no subset of cards of the same color has the sum of the numbers equal to . Find the greatest such that Egor will not be able to paint the cards in such a way.
that has 8097 distinct points with area of a triangle that has 3 points belong to all 
. Prove that there exists a triangle 
that has the area 
contains at least 2025 points that belong to ( each of that 2025 points can be inside the triangle or lie on the edge of triangle )X
is even it's contradiction. I mean 
.
and 
,we can see the only solution is
and it is contradiction.
and 
. So 
.İt means 
.
but 
.İt is contradiction.
we have 2 option. Firstly 
but 
is contradiction. Secondly 
but is contradiction. I mean 
. So 
. If 
but is contradiction. I mean 
. The only solution is 
.
be a closed convex set that contains 
but no other point with integer coordinates. Suppose that 
, the area of , is equally distributed among the four quadrants. Prove that 
of positive integers, a partition of into two disjoint nonempty subsets 
and 
is 
if the least common multiple of the elements in is equal to the greatest common divisor of the elements in . Determine the minimum value of such that there exists a set of positive integers with exactly 
good partitions.
are lengths of sides of some triangle. Prove the inequality
where 
. Point 
is the circumcenter of triangle , and 
is the projection of point 
onto line 
. The midpoints of , 
, and 
are 
, 
, and 
, respectively. The line 
intersects 
and 
at points 
and 
, respectively. Prove that is the incenter of triangle 
.
and 
Prove that
id defined by![\[
f(x) = \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
\]](http://latex.artofproblemsolving.com/0/2/c/02c7b4af3cac77da8e3cc102c79a7db2e95f437f.png)
now what is CDF or cumulative distribution function ![$F_X(x)=P[X\leq x]$](http://latex.artofproblemsolving.com/7/9/1/79150c7527581feec873c9c790f8aeddc8f8efb8.png)
.
,
.
.
where 
is the CDF of the standard normal distribution.