[/quote]
,
.
be an acute-angled triangle in which 
and 
.
and point 
such that 
,
and 
.
be the circumcenter of triangle 
,
the orthocenter of triangle 
the point of intersection of the lines 
and 
.
consecutive even numbers, such that the square of their product divides the sum of their squares.
be a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
and 
are the feet of the altitudes through 
and 
respectively. The incenters of the triangles 
and 
are 
and 
respectively; the circumcenters of the triangles 
and 
are 
and 
respectively. Prove that 
and 
are parallel.
with 
prime and ![\[a^p+b^p=p!.\]](http://latex.artofproblemsolving.com/f/e/b/feb27894b1f6497559eeb094cfe92256bc26688e.png)
such that 
.
such that 
.
.
be a cyclic quadrilateral with circumcenter 
,
is not a diameter of its circumcircle. The lines 
and 
lies between 
and 
lies between 
is acute and let 
and 
and 
.
,
.
which satisfy the condition![\[ f(m + n) \geq f(m) + f(f(n)) - 1
\]](http://latex.artofproblemsolving.com/4/b/d/4bddce124cf0da82cb8bb6c6ae2325239354f27c.png)
Find all possible values of 
be a positive integer and 
be a positive integer coprime to 
.
,
,
Find, in terms of 
,
such that 
is divisible by 
.
table are initially white. Alice and Bob play a game. First Alice paints 
Y by bryanguo, peppapig_, centslordm
Consider the set of all points 
such that their isogonal conjugate 
in 
has 
parallel to 
.
Prove that the midpoint of all
in this set lies on a rectangular hyperbola, and locate its center.
such that their isogonal conjugate 
in 
has 
parallel to 
.Prove that the midpoint of all
in this set lies on a rectangular hyperbola, and locate its center.This post has been edited 2 times. Last edited by qwerty123456asdfgzxcvb, Oct 19, 2024, 6:20 AM
and 
.
.
,
is real, so 
.
gives
This hyperbola is clearly rectangular. Rewrite this as
,
.
,
(with a weird limiting case where 
is the 
and the cubic at 
(circle points). Then we have 
which only happens if 
and 
are perpendicular.)
be a line through it. Let 
.
,
is harmonic and thus 
,
be the isogonal conjugate WRT 
.
,
,
,
respectively. One can see that 
and 
.
so triangles 
and 
are similar, and similarly we have 
and 
are similar. Thus we obtain the equalities 
and 
.
and 
one can solve a linear system of equations for 
and 
,
is the point 
.
.
and 
by observing that 
thus we know 
.
,
which is just 
