[/quote]
,
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
with real coefficients of degree 
such that 
divides 
.
such that:
.
-
.
so that any 
.
satisfying the following:
;
and 
;
;
has all of its coefficients being even.
and 
Prove that
points on the plane with diameter 
and two not necessarily distinct points 
not equal to 
or 
satisfying that![\[|AX-XY|+|BY-XY|\leq \frac{2D}{N-2}.\]](http://latex.artofproblemsolving.com/f/3/6/f3649872c33e1db8880b4aedace38d9439f2b4ae.png)
.
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
of positive integers for which there exist positive integers 
and 
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
such that 
for all 
?
and 
Prove that
Y by Mango247, Mango247, Mango247
In an acute-angled triangle 
point 
is midpoint of the side 
, points 
and 
are projections of point on the sides 
and 
respectively. The circles circumscribed around triangles 
and 
, intersect at the point 
different from 
. Prove that lines 
and are perpendicular.
point 
is midpoint of the side 
, points 
and 
are projections of point on the sides 
and 
respectively. The circles circumscribed around triangles 
and 
, intersect at the point 
different from 
. Prove that lines 
and are perpendicular.This post has been edited 1 time. Last edited by parmenides51, Sep 20, 2020, 11:48 PM
