[/quote]
,
such that for all 
the followings hold:
be a triangle with circumcentre 
.
lie in the interiors of the sides 
respectively, such that 
is perpendicular to 
and 
is perpendicular to 
.
be the circumcentre of triangle 
.
and 
?
of positive integers, such that every sum
(with 
)
for all 
;
for all 
?
denotes the greatest common divisor of 
,
.
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](http://latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
,
,
be positive real numbers such that 
.![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
and 
and 
Prove that![\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]](http://latex.artofproblemsolving.com/0/a/b/0ab4b4e6c28a1ae67788525642661dbcd04d0bd8.png)
and in 
steps and has to reach 
is valid but 
is not valid.
,
.
be the midpoints of 
and 
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
such for any 
.
satisfy,
Where 
}
be an integer. We are given a balance and 
weights of weight 
.
Y by
Let 
be an acute triangle. The bisector of 
intersects side 
in 
. The circumcircle of triangle 
intersects side 
in 
and 
with 
. The line parallel to 
which passes through 
intersects line 
in 
. Prove that the triangle 
is isosceles.
be an acute triangle. The bisector of 
intersects side 
in 
. The circumcircle of triangle 
intersects side 
in 
and 
with 
. The line parallel to 
which passes through 
intersects line 
in 
. Prove that the triangle 
is isosceles.
,
is cyclic and 
.
