[/quote]
,
such that for all 
the followings hold:
.
lie in the interiors of the sides 
respectively, such that 
is perpendicular to 
and 
is perpendicular to 
.
lies on the line 
and 
on that line.)
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 
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 Mango247, Mango247, Mango247
Let 
be feet of altitudes from vertices 
of triangle 
, respectively. Line parallel to 
passing through 
intersects line 
at point 
. Point 
is the midpoint of segment 
. Show that 
.
be feet of altitudes from vertices 
of triangle 
, respectively. Line parallel to 
passing through 
intersects line 
at point 
. Point 
is the midpoint of segment 
. Show that 
.This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 2:36 AM
C
B
.
is cyclic or 
.
or 
is cyclic. So, 
.
