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.
such that 
be nonnegative real numbers such that 
. Find min
, 
, 
cut three common chords concurrent at 
. Consider two distinct circles 
, 
which are internally tangent to all 
. Determine, with proof, which of the following two statements is true. is the insimilicenter of and is the exsimilicenter of and .
Prove that
Proposed by Phan Ngoc Chau
be a triangle, and let 
be a point on side 
. A line through intersects side 
at and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle again at point 
distinct from point 
. The lines 
and 
intersect again at 
and 
respectively.
grid has had the central unit square removed. For which positive integers 
is it possible to cut the remaining area into 
and 
rectangles?
is a prime number and 
are positive integers. Prove that there exist infinitely 
such that 
are divisible by 
![\[\varphi(n)=n\prod_{p\mid n} {\left( 1-\frac{1}{p} \right)}\]](http://latex.artofproblemsolving.com/f/c/0/fc0c4223b31182c4bc677099096a76648cd79cf4.png)
where the product is taken over the primes .
are distinct primes dividing such that 
for every 
, then divides 
. there are an infinite number of primes 
(this special case of Dirichlet's theorem is also susceptible to an elegant elementary proof via cyclotomic polynomials).
pairwise disjoint groups of 
distinct primes, each being 
. Let 
denote the products of the primes in their respective groups. Now the result follows via the Chinese Remainder Theorem by choosing positive integers such that![\[a+i\equiv 1 \pmod{ N_{i+1}}\]](http://latex.artofproblemsolving.com/2/0/e/20e2354a7f15403ab180f02e7311c0756a9895ae.png)
for every 
.
and 
Prove that
Where 
such that 
