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.
be real numbers such that 
. Find the value of 
be real numbers such that 
and 
. Find the value of 
with 
-excenter 
, the foot of the perpendicular from to 
is 
. Let the midpoint of segment 
be 
, 
lies on arc (not containing ) satisfying 
, 
intersects the circumcircle of at 
. If 
and intersect at 
, the perpendicular bisector of 
intersects 
at 
respectively, prove that 
are concurrent.
such that
be a positive integer larger than 
. Build an infinite set 
of subsets of 
having the following properties: distinct sets of have exactly one common element;
distinct sets of have void intersection.
and 
be positive integers such that 
divides 
. Prove that 
.
where 
are nine distinct integers. Prove that there exists an integer 
such that for all integers 
the number 
is divisible by a prime number greater than 20.
be the lengths of the sides of a triangle. Prove that![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](http://latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
of points in the plane is balanced if, for any two different points and 
in , there is a point 
in such that 
. We say that is centre-free if for any three different points , and in , there is no points 
in such that 
.
, there exists a balanced set consisting of 
points. for which there exists a balanced centre-free set consisting of points.
let 
be the number whose 
-th digit after the decimal point is the 
-th digit after the decimal point of 
. Show that if is rational then so is 
.
from the set of positive integers into itself is defined by the equality ![\[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]](http://latex.artofproblemsolving.com/2/a/5/2a5a20395450d37d0352ab75e9e4875ab7803805.png)
for every two relatively prime 
.
the equation 
has a solution.
such that the equation has a unique solution.
be an even integer and let 
be real numbers satisfying 
.
such that 
be a prime number, and 
be a sequence defined by 
and 
, 
. Show that one of the first 
terms of the sequence is not prime.
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
.
Then, by Fermat 