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.
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
be an acute triangle with orthocentre 
and circumcircle 
. Let 
be the point
on . The line through , parallel to 
, intersects 
and 
at 
, respectively. Let 
intersect the circumcircle of triangle 
again at 
. Let the tangent to at intersect 
at 
. Prove that lines 
and 
intersect on .
numbers on circle, and no one number is divided by other. In same time for all numbers we make next operation:
are two neighbors (
is left neighbor) , then we write between 
number 
and erase 
we can receive ?
numbers 
then after operation we receive circle with numbers 
.
. Prove that![\[
\sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2}
\]](http://latex.artofproblemsolving.com/9/0/3/903220c471bdd3028dda2fcb13874a4bcf752872.png)
without using the Rearrangement Inequality or Chebyshev's Inequality.
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
, such that 
. Prove that
.
such that for all positive real numbers 
and 
:
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)
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)
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
Let and is the sum of square numbers. Prove that: is the sum of square numbers
Let and is the sum of square numbers. Prove that: is the sum of square numbers
be equal to divided by the largest power of 
that divides . Legendre's three-square theorem says that is expressible as the sum of three squares iff is not 
mod 
.
is never mod , so even 
always works. If is odd, if 
is mod , then 
is even and thus 
is even. Thus 
is odd. Observe that if is odd then 
. Thus if is mod then 
is mod . However, any odd number to an odd power is congruent to itself mod , so this means 
, contradiction.
, which would make it trivial.
?
are non-negative real numbers and 
Prove that
the equality hold for 
or 
or any
be positive real numbers. Prove 
Prove that
