[/quote]
,
such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
,
be the number of all positive divisors of 
.
of real numbers that satisfy the system of equations![\[ \begin{cases}x^3 = 3x-12y+50, \\ y^3 = 12y+3z-2, \\ z^3 = 27z + 27x. \end{cases}\]](http://latex.artofproblemsolving.com/d/4/d/d4d6c485b4653577e732ff235448bb779914a576.png)
be an acute triangle with orthocenter 
,
be a point on the side 
,
and 
.
and 
are the feet of the altitudes from 
is the circumcircle of 
,
be the point on 
is a diameter of 
the circumcircle of triangle 
,
be the point such that 
is a diameter of 
and 
such that for every 
,
,
equals the number of times 
be the sequence 
so that 
for each 
and so that the decimal number 
is divisible by 
for each 
.
is irrational.
points on the plane, no three of which are collinear, and some set of line segments between them. We say that a subset of these line segments is called a "pairing" if every one of these 
,
distinct (but not necessarily disjoint) pairings.
be a polynomial with real coefficients such that 
for all 
.
for some real constant 
and 
.
where 
.
be polygons, no two of which are similar. Show that there are polygons 
where 
is similar to 
so that for no 
does 
.![\[ 2^n -2, \;\; 3^n -3, \;\; 4^n -4, \;\; 5^n-5, \, \ldots \ldots \]](http://latex.artofproblemsolving.com/d/b/1/db17ef301bf3fd01f8b310f91a14b9dee045e11f.png)
is 
.
?
be a function such that 
.
.
as its circumcentre, 
as its nine-point centre.
,
and 
to be the midpoint of the smaller arcs.
as the isogonal conjugate of the Nagel point (i.e. the exsimillicenter of the incircle and circumcircle)
as the ninja point of 
as the ninja point of the contact triangle
Points 
are collinear, that is 
Points 
and 
.![$S=\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9}$](http://latex.artofproblemsolving.com/9/d/9/9d93ff9d9f7355e452f1007aeb87423e3491353a.png)
Y by Adventure10, Mango247, and 2 other users
Let 
be nonnegative real numbers, no two of which are zero. Prove that
![\[ (ab+bc+ca)\left(\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2}+\frac{1}{(a+b)^2}\right) \ge \frac{9}{4}\]](//latex.artofproblemsolving.com/5/7/b/57b789b81eef052f6978eee67766e314c9ef1e21.png)
This is not a new inequality with someone-Iran TST 1996. And now, proving it is not hard as the past but have you think of find more solutions for it? I think it will be a very interesting work, and luckily, I found 10 proofs for it. In here, I wanna see more solutions from you. Thanks
P/s: I expect that the solution dont use SS,SOS, pqr ...
be nonnegative real numbers, no two of which are zero. Prove that![\[ (ab+bc+ca)\left(\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2}+\frac{1}{(a+b)^2}\right) \ge \frac{9}{4}\]](http://latex.artofproblemsolving.com/5/7/b/57b789b81eef052f6978eee67766e314c9ef1e21.png)
This is not a new inequality with someone-Iran TST 1996. And now, proving it is not hard as the past but have you think of find more solutions for it? I think it will be a very interesting work, and luckily, I found 10 proofs for it. In here, I wanna see more solutions from you. Thanks
P/s: I expect that the solution dont use SS,SOS, pqr ...
I don't know your solution for it.
![\[ (ab + bc + ca)\left(\frac {1}{(b + c)^2} + \frac {1}{(c + a)^2} + \frac {1}{(a + b)^2}\right) \ge \frac {9}{4}
\]](http://latex.artofproblemsolving.com/8/d/c/8dcbe92a5c1c2445a4fc72b72cefbf559212e026.png)
![$ ([5,1,0]-[4,2,0])+3([5,1,0]-[3,3,0])+([4,1,1]+[2,2,2]-2[3,2,1]) \ge 0$](http://latex.artofproblemsolving.com/1/2/1/121991122bbbf636859a1df3d119d27af82a9e13.png)
multiplied with 
![$ 4(a + b)(b + c)(c + a)[abc - (a + b - c)(b + c - a)(c + a - b) ] \ge (a - b)^2(b - c)^2(c - a)^2$](http://latex.artofproblemsolving.com/1/4/d/14d9d2ebfa663a322c8373cd4f93741f2a95becc.png)
Which is true by am-gm
i
