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.
is given, 
, 
are graphs on the same set of 
nodes, having 
connected components respectively. Prove that 
has at least 
connected components.
and 
intersect at points and . A line through intersects and at points 
and 
, respectively. Line 
intersects at point 
, and line 
intersects at point 
. If 
is the circumcenter of 
, prove that 
.
be a positive integer. Initially the sequence 
( times) is written on the board. In each round, Ananya choses an integer 
and a subset of the numbers written on the board and adds to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing?
be real numbers . Prove that : There exist positive integer 
such that 
Where 
, 
, 
are pairwise distinct real numbers satisfying 
Prove that
Let , , are pairwise distinct real numbers satisfying 
Prove that
and 
be positive integers. is a divisor of 
and is a divisor of 
.
.
and 
be the two foci of ellipse 
. 
is a point on . Let be the center of the excircle of 
. When 
, find the minimum eccentricity of .
of natural numbers with elements with odd number. A nonempty subset 
of is called 
if the product of the elements of divisible by the sum of the elements of , but not by its square. It is known that the set itself is good. Determine the maximum number of subsets (including itself).
be positive reals with sum 1. Show that for any -tuple of reals 
, there exists an -tuple of reals 
satisfying 
for all 
, where 
.
and 
Prove that 
be real numbers such that 
. Find the minimum of 
and 
Prove that 
positive integer arrays 
, and they are all infinity arrays.
, that for every integer 
![\[
\sqrt[3]{\sqrt[3]{x - \frac{3}{8}} - \frac{3}{8}} = x^3 + \frac{3}{8}
\]](http://latex.artofproblemsolving.com/b/b/9/bb99cd588b94b5716738e6e9ea01d15377b6efef.png)
has exactly two real positive solutions 
and 
. Compute 
.
of the pyramid 
are equal to 
, and 
, respectively the base of the ABC is a regular triangle. It is known that the area of the lateral faces of the pyramid are equal to each other. Find the volume of the pyramid .
of the pyramid are equal to , and , respectively the base of the ABC is a regular triangle. It is known that the area of the lateral faces of the pyramid are equal to each other. Find the volume of the pyramid .
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
and 
Prove that
has length 
.
has sides 
,
has sides 
,
for convenience), we have 
,
,
.
,
.
times the square of the volume) of this purported tetrahedron to check if it positive. However, 
)