[/quote]
,
and 
Prove that
Where 
and 
be positive integers with 
.
,
.
be the orthocenter of acute triangle 
,
be the foot of the altitude from 
to 
,
.
intersects line 
and 
.
.
such that 
is an integer for every positive integer 
and ![$P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/c/6/4/c64176e52d94032e571b4579e35f9e6adfde675a.png)
such that 
.
such that every equation: 
has no real roots
.
and 
are on 
respectively such that 
.
and 
meet at 
is the incenter of 
and 
.
and 
are circles with diameter 
and 
,
such that 
.
.
points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?
be an integer, and let 
be positive real numbers such that 
.![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](http://latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
not be a trapezoid such that there is a circle centered at 
,
,
,
.
,
be the circumcenters of the triangles 
,
,
,
,
,
,
,
.
satisfies 
.
.
Y by
[![Triangle with angles 40°, 30°, and 10°, and AB = EC][1][1](https://imgur.com/a/6oIVAbI )]!
I came across this geometry problem while practicing triangle properties. It involves given angles and an equality between two segments: .
I have tried using the angle sum property in triangles and the isosceles triangle theorem (since ), but I am unsure how to proceed to find . I also considered exterior angles but could not establish a clear relationship.
I find this problem interesting because it seems to involve multiple geometric properties, and I would like to understand the best approach to solve it. Any hints or explanations would be greatly appreciated!
I came across this geometry problem while practicing triangle properties. It involves given angles and an equality between two segments: .
I have tried using the angle sum property in triangles and the isosceles triangle theorem (since ), but I am unsure how to proceed to find . I also considered exterior angles but could not establish a clear relationship.
I find this problem interesting because it seems to involve multiple geometric properties, and I would like to understand the best approach to solve it. Any hints or explanations would be greatly appreciated!
