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.
angle 
is greater than
. Let 
is midpoint of 
. 
and 
are the feet of the altitude from and respectively. 
and 
are midpoint of 
and 
respectively. If 
intersect the line through 
parallel to in 
, prove that 
.
, 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 
:
is natural and 
be a prime number. Prove that there exists a natural number 
such that![\[
p \mid m^n - n.
\]](http://latex.artofproblemsolving.com/3/a/3/3a38b299f0328d176286d65e02b92e9918e8109d.png)
just pick 
, otherwise 
is coprime to 
. By Chinese Remainder Theorem there exists an 
such that 
and 
, pick this , we see that both 
and leave remainder modulo , so we are done.
?
are non-negative real numbers and 
Prove that
the equality hold for 
or 
or any
be positive real numbers. Prove 
Prove that
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)
