AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
has two isogonal conjugate points 
and 
. The circle 
intersects circle 
at 
, and the circle 
intersects circle 
at 
. Prove that 
and 
are isogonal conjugates in triangle . is the circle with diameter 
, Circle is the circle with diameter 
.
be a real or complex inner product space. Suppose that 
and that 
Then
, s.t :
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 
. Denote by 
and 
the midpoints of 
and 
. Rays 
and 
meet 
and 
at and 
, respectively. Prove that there exists a point 
, lying outside quadrilateral , such that
bisects both angles 
, 
, and
.
be positive real numbers such that 
Find the minimum value of 
lies on the bisector of an acute angle with vertex . Let , be the projections of to the sidelines of the angle. The circle centered at with radius 
meets the sidelines at points 
and 
such that 
. Prove that the circle with center touching 
and the circle with center touching 
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 .
on the whiteboard. In each move, she selects two distinct numbers, 
and 
, erases them, and replaces them with 
. She repeats this process until only one number, 
, remains. What are all the possible values of ?
, and the sum of all numbers decreases by 
as well. This value is equal to 
, and it will always be equal to 
. We add to get the sum of all numbers when there is number, and the only possible value of is 
.
satisfying
Prove
satisfying
Prove
Prove that
and 
Prove that
Prove that
Where 
