[/quote]
,
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
with incenter 
.
at 
.
and 
be points on the side BC such that 
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
,
,
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
be the set of positive real numbers. Find all functions 
such that, for all 
,
be a cyclic quadrilateral. The point 
is the reflection of 
the intersection of 
is the reflection of 
.
be the reflection of 
.
concyclic.
. Prove that: 
Y by NumberX, Adventure10, Mango247
I have realized that I hate how the US way of getting into IMO because I really suck at computation problems. Other countries begin with proof based ones and stick with that, but US does computation and then proof. I suppose it's good for easy grading since there are a bunch of people taking the first level and lesser as you go up. However I absolutely suck at these.
Are there any alternate US competitions that are only divided by grade level? Eg. Tournament of Towns (but that's Russian or something) is just one thing, all proof based, divided by grade level. Anything like that in the US?
Are there any alternate US competitions that are only divided by grade level? Eg. Tournament of Towns (but that's Russian or something) is just one thing, all proof based, divided by grade level. Anything like that in the US?
.
