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.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
such that if 
and 
are relatively prime divisors of 
, then 
divides .
such that:
such that:
be an even prime number and a sequence 
satisfy 
and 
Prove that always exists positive integer 
satisfying for all positive integers 
, then 
.
is "not congruence"
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 
.
. Points 
and 
are symmetric to its vertices with respect to opposite sides. 
is the intersection point of lines 
and 
. Points
and 
are defined similarly. Prove that the lines 
and 
are parallel.
. Points and are symmetric to its vertices with respect to opposite sides. is the intersection point of lines and . Points and are defined similarly. Prove that the lines and are parallel.
.
, are the external bisectors of the triangle 
, so 
- bisectors of 
.
is cyclic. But 
so lie on the bisector of . Sum up, 
are collinear.
, 
and 
- point of intersection of 
and 
with the circumcircle . Well known 
and 
.
(because 
and they are isosceles)
(because 
and 
)![\[ \frac{IO}{CH}\ = \frac{OB}{CB_3}\ = \frac{C_2O}{CC_3} \]](http://latex.artofproblemsolving.com/e/4/4/e44758cf328f9687422cf46625382abb208e24d8.png)
But 
= 
, so![\[ \frac{C_1H}{CH}\ = \frac{C_2O}{CO} \]](http://latex.artofproblemsolving.com/6/4/0/64098ce9f366b482121e972ab2479508b042055a.png)
It is means that and 
are parallel.
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
