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 for all positive real numbers 
and 
:
be positive real numbers such that 
. Show that![\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]](http://latex.artofproblemsolving.com/e/a/1/ea1f56cc594862673040411c3ad322cb00cfe904.png)
inscribed in 
with 
being the midpoint of 
. The tangents at 
of intersect at 
. Let 
be the projection of 
onto 
. On the perpendicular bisector of , take a point 
that is not on and different from M. Circle 
intersects 
at 
. Lines 
and 
intersect at 
. Prove that 
is an isosceles triangle.
[hide=P#]a1b2c3[/hide] [hide=S#]a1b2c3[/hide]
can be defined as a sequence such that 
, and 
, such that 
. The function 
is such that 
. How many numbers within the interval 
for the function 
are perfect squares?
be the set of all positive integers. We say that a function 
is Georgian if 
and, for every positive integer 
, there exists a positive integer 
such that![\[
f^{(k)}(n) = 1, \quad \text{where } f^{(k)} = f \circ f \cdots \circ f \quad \text{(applied } k \text{ times)}.
\]](http://latex.artofproblemsolving.com/5/3/c/53ca7c663237d939a7a049aa3569f8d595b259cf.png)
If 
is a Georgian function, we define, for each positive integer , 
as the smallest positive integer 
such that 
. Determine all positive real numbers 
for which there exists a Georgian function such that, for every positive integer 
, it holds that 
.
prove that
denotes the sum of the integers, 
denotes the sum of the square of the integers and 
denotes the sum of cubes of the integers(that is, if the three integers are 
, then 
, 
, 
). If 
and 
, find 
.
and 
that satisfy the system of equations:
and 
be the sets of integers and rationals respectively. into three non-empty subsets 
such that the sets 
are disjoint? into three non-empty subsets such that the sets are disjoint?
denotes the set 
, for 
and for 
.
). We have 
distribute the 
factors of 
among the 
values. Continuing for all primes gives the product 
. For the example, the answer is 
. The number of unordered pairs would be messy because you have to do casework with pairs containing repeated values.
, how many pairs of 
numbers multiply to 
?
and 
.![\[ \frac{a^2+1}{a+1}+\frac{b^2+1}{b+1}+\frac{c^2+1}{c+1} \geq 3\]](http://latex.artofproblemsolving.com/7/d/2/7d2e34116452912cd3da2925312ad500bb976f15.png)
:o
so 
and since 
is just 
therefore 
so 
,
numbers that multiply to 
,
,
then their product far exceed 
)
I think?