[/quote]
,
satisfying
Prove
has two isogonal conjugate points 
and 
.
intersects circle 
at 
,
intersects circle 
at 
.
and 
are isogonal conjugates in triangle 
,
.
satisfying
Prove
be a real or complex inner product space. Suppose that 
and that 
Then
,
such that 
,![\[
f(x^2+f(y))=f(xf(x))+y
\]](http://latex.artofproblemsolving.com/d/e/d/dedd2338779fcc6723077ce586713fb4a68b9931.png)
be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at 
.
and 
the midpoints of 
and 
.
and 
meet 
and 
at 
,
,
bisects both angles 
,
,
.
Prove that
and 
Prove that
be positive real numbers such that 
Find the minimum value of 
lies on the bisector of an acute angle with vertex 
meets the sidelines at points 
and 
such that 
.
and the circle with center 
are tangent.
are nonzero real numbers such that![\[
\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b},
\]](http://latex.artofproblemsolving.com/6/d/9/6d9aabd8130b99d7128de9a3f063eb4c7124febf.png)
let 
be the maximum value of the expression![\[
\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}.
\]](http://latex.artofproblemsolving.com/a/b/2/ab28769126365574d4c0108500c370826082de6b.png)
Determine the sum of the numerator and denominator of the simplified fraction representing 
Prove that
Where 
Y by buratinogigle
Let 
be an acute-angled, non-isosceles triangle with circumcenter 
and incenter 
, such that
![\[
\prod_{\text{cyc}} \left( \frac{1}{a+b-c} + \frac{1}{a+c-b} - \frac{2}{b+c-a} \right) \neq 0,
\]](//latex.artofproblemsolving.com/7/8/1/78125759bf5c73149e4a56538e9b0c44309704a7.png)
where
, 
, and 
.
Let
, 
, and 
be the excenters opposite to vertices 
, 
, and 
, respectively, and let 
, 
, and 
be the Lemoine points of triangles 
, 
, and 
, respectively.
Prove that the circles
, 
, and 
all pass through a common point 
. Moreover, the isogonal conjugate of with respect to lies on the line 
.
be an acute-angled, non-isosceles triangle with circumcenter 
and incenter 
, such that![\[
\prod_{\text{cyc}} \left( \frac{1}{a+b-c} + \frac{1}{a+c-b} - \frac{2}{b+c-a} \right) \neq 0,
\]](http://latex.artofproblemsolving.com/7/8/1/78125759bf5c73149e4a56538e9b0c44309704a7.png)
where
, 
, and 
.Let
, 
, and 
be the excenters opposite to vertices 
, 
, and 
, respectively, and let 
, 
, and 
be the Lemoine points of triangles 
, 
, and 
, respectively.Prove that the circles
, 
, and 
all pass through a common point 
. Moreover, the isogonal conjugate of with respect to lies on the line 
.Attachments:
This post has been edited 1 time. Last edited by Ktoan07, Apr 18, 2025, 10:26 AM
