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.
, point 
is the midpoint of 
, 
the point of intersection of the tangents at points 
and 
of the circumscribed circle of , 
is the midpoint of the segment 
. The segment 
meets the circumcircle at the point 
. Prove that 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
such that 
+1 w
that satisfy the equation
with incenter 
. The incircle of triangle is tangent to at 
. Let and be points on the side BC such that 
and 
, respectively. Let 
and 
be the incenter of triangles 
and 
, respectively. Prove that 
is the Euler line of triangle 
.
, , 
, 
, where and 
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
be reals such that 
and 
Prove that
in 
colors, there exists an arithmetic progression of size 
. How can we derive Van der Waerden's theorem for 
from this?
and Szemeredi's theorem. But this is, obviously, not the intended solution and so I wonder if there exists a solution which uses nothing but relatively elementary methods (maybe even the same induction on , or on ). Thank you.
be a countable set and let A be some set of finite subsets of X.
, A contains a monochromatic set.
, this immediately implies Van der Waerden.
, there 
a -coloring 
of 
such that 
doesn't contain monochromatic set. Then by Dirichlet, there must be infinitely many colorings 
, such that 
are all colored in the same way. If repeat the argument for the 
, then there are infinitely many 
, so that the 
are all colored in the same way. And so on. We will be gradually constructing the coloring of 
, call it , where for any prefix 
there is no monochromatic set in if we only color this prefix. By the assumption, contains some monochromatic set 
. But that means if we only color prefix 
there exists a monochromatic set of , which is a contradiction to the construction of 
:
Where 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
