by 
and the number of distinct prime divisors of 
.
be a positive integer. Prove that there exist infinitely many positive integers 
such that 
and 
does not divide 
for any positive integers 
satisfying 
.
such that for any 
:
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
such that 
for all 
Y by khan.academy, cubres, Mathdreams
Consider a triangle 
and some point 
on 
. The perpendicular from to 
intersects the circumcircle of 
at 
and the perpendicular from to 
intersects the circumcircle of 
at 
. Show that the line 
does not depend on the choice of .
(Shining Sun, USA)
and some point 
on 
. The perpendicular from to 
intersects the circumcircle of 
at 
and the perpendicular from to 
intersects the circumcircle of 
at 
. Show that the line 
does not depend on the choice of .(Shining Sun, USA)
This post has been edited 6 times. Last edited by Bata325, Apr 13, 2025, 2:54 AM
Reason: italics
Reason: italics
denote the circumcenter of 
.
.
denote the foot of the altitude from ![\[\angle{BAO} = \frac{180^{\circ} - \angle{AOB}}{2} = \frac{180^{\circ} - 2\angle{C}}{2} = 90^{\circ} - \angle{C} = \angle{HXC} = 180^{\circ} - \angle{BXH} = \angle{BAQ}.\]](http://latex.artofproblemsolving.com/e/2/d/e2d9375826c14c5e7197102e8c1d14f2dad41068.png)
Hence, the points 
are collinear. Similarly 
are all collinear.
,
be the midpoint of 
hence 
are the centres of the circles 
respectively
and 
such that 
and 
and 
are the perpendicular bisectors of the radial axes 
and 
hence 
and 
so 
and 
and the required claim is proved
![\[\angle{BAO} = \frac{180^{\circ} - \angle{AOB}}{2} = \frac{180^{\circ} - \angle{C}}{2} = 90^{\circ} - \angle{C} = \angle{HXC} = 180^{\circ} - \angle{BXH} = \angle{BAQ}.\]](http://latex.artofproblemsolving.com/7/d/9/7d972f3079ea48f9b57a8f171777f9dabada44e0.png)
Hence, the points 
hence 
where ![\[\measuredangle BAQ = \measuredangle CXQ = 90 + \measuredangle BCA\]](http://latex.artofproblemsolving.com/9/4/2/9429375d4f0c28c6a6939e228bc971645d77e8ec.png)
which implies that ![\[\measuredangle CAQ = \measuredangle CXP = 90 + \measuredangle CBA\]](http://latex.artofproblemsolving.com/0/0/a/00ab53c14582f60a8519bd3daef06f3048629879.png)
which implies that 
