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.
+1 w
and 
. Prove that:![\[ \frac{a^2+1}{a+1}+\frac{b^2+1}{b+1}+\frac{c^2+1}{c+1} \geq 3\]](http://latex.artofproblemsolving.com/7/d/2/7d2e34116452912cd3da2925312ad500bb976f15.png)
+1 w
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.
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 
.
grid (
), we color some of its cells black.A coloring is called balanced if each row and each cell contains exactly 
black cells.Detemine the number of balanced colorings.
be a quadrilateral inscribed in a circle with center and be the intersection of segments 
and 
. Let 
be the circumcircle of 
and 
be the circumcircle of 
. The tangent to at and the tangent to at meet at 
. The tangent to at and the tangent to at meet at 
. Show that 
.
, let 
denote the number of 
s in the binary representation of . Prove that for any positive integer ,![\[\sum_{i=1}^{n}(-1)^{s(3i)} > 0.\]](http://latex.artofproblemsolving.com/d/e/9/de975cc2632685d7a3f71c023115535d0aeb5dbb.png)
Holden Mui
for which there exist real numbers 
satisfying 
, 
and
for 
.
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers .
. such that 
, then by induction it follows that 
, which forms a valid sequence.
. Then, we have:
Hence, we obtain
, where 
, it follows that 
. For sufficiently large , the left-hand side grows linearly while the right-hand side grows quadratically:
which is a contradiction.
.
?
