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 
are the lengths of a right triangle.
be positive real numbers such that![\[
x + y + z = 3.
\]](http://latex.artofproblemsolving.com/1/c/6/1c6c52fbb7220d19c20a6f543c87b8eaa6bdf9aa.png)
Find the minimum value of![\[
x^2 + y^2 + z^3
\]](http://latex.artofproblemsolving.com/b/1/6/b1618ed295b6c950115cbec15f882cb4f9f0fcfa.png)
and express it in the form 
, where 
. Compute 
.
be a positive integer that is not divisible by 
. Furthermore, let 
be a positive integer. Prove that none of the numbers of the form 
is a square number.
+2 w
such that is a multiple of 
and has exactly positive factors.
be real numbers such that 
Find the maximum value of the expression 
different bus lines in a city, each stopping at exactly 
stations and running in both directions. Nevertheless, for every two different stations there is always a bus line connecting these two stations. Determine the maximum number of stations in this city.![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x^2 + y) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/c/7/c/c7cd7c9a4d4372afc23e4503bb1c05fc7d07e788.png)
such that:
with all 
is real number
be an isosceles triangle with 
and circumcircle 
. The line through 
perpendicular to 
is denoted by 
. Furthermore, let 
be any point on . The circle 
with center and radius 
intersects 
once more at point 
and the circumcircle once more at point 
. Prove that the points 
and 
lie on a straight line.
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
be an acute triangle. Points 
, and lie on a line in this order and satisfy 
. Let and 
be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points and lie on lines and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
+1 w
, find the number of terms with coefficient divisible by 
.
into sets of elements. Other then the fact that is a divisor of 
and that this chapter is constructive counting, this is not intuitive at all. Am i missing something?
into sets of elements. Other then the fact that is a divisor of and that this chapter is constructive counting, this is not intuitive at all. Am i missing something?
into sets of elements. Other then the fact that is a divisor of and that this chapter is constructive counting, this is not intuitive at all. Am i missing something?
specifies the greatest common divisor of 
and 
.
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
