[/quote]
,
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 
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.
,
lie on a line in this order and satisfy 
.
and 
,
is acute, and let 
be its orthocentre. Points 
lie on lines 
and 
and 
and 
and 
such for any 
.
satisfy,
Where 
}
be an integer. We are given a balance and 
weights of weight 
.
to satisfy that the equation 
has at least 1 roots 
of the equation![\[n^3=p^2-p-1\]](http://latex.artofproblemsolving.com/f/f/8/ff88093bb0aa93afcfdff3ae25149ec5fbacb61f.png)
is a prime number and 
?
Y by
Let 
be the center of the circumcircle of an acute non-isosceles triangle 
. The lines 
and 
intersect the sides 
and 
at points 
and 
, respectively. On the sides and are taken such different from and points 
and 
, respectively, that 
and 
. A line parallel to 
is drawn through the point , and a line parallel to 
is drawn through the point , and we denote by 
the intersection point of these lines. Prove that the radii of the circumcircles of the triangles 
and 
are equal.
IP Nagel
be the center of the circumcircle of an acute non-isosceles triangle 
. The lines 
and 
intersect the sides 
and 
at points 
and 
, respectively. On the sides and are taken such different from and points 
and 
, respectively, that 
and 
. A line parallel to 
is drawn through the point , and a line parallel to 
is drawn through the point , and we denote by 
the intersection point of these lines. Prove that the radii of the circumcircles of the triangles 
and 
are equal.IP Nagel
This post has been edited 4 times. Last edited by parmenides51, Jun 21, 2022, 1:54 AM
