AoPS Academy
be circumcenter of a non-isosceles triangle 
and 
be a point in the interior of 
. Let 
be foots of perpendicular lines from to 
. Suppose that 
is cyclic and 
is the circumcenter of , 
. Prove that 
bisects 
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](http://latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
be a triangle with 
. Let 
be the midpoint of 
, and let 
be a point on segment such that 
. Let 
be the point of intersection, different from 
, of the circumcircle of triangle 
and line 
. Define 
and 
as the points of intersection of line 
with 
and 
, respectively. Prove that is the midpoint of 
.
have orthocenter and altitude 
with 
on side 
. Let be the midpoint of side , and let 
be the reflection of over . Let 
be a point on line 
such that lines 
and are parallel, and let the circumcircles of 
and 
meet again at 
. Prove that 
.
students and their test results on 
different subjects, we say that student 
dominates student 
if and only if outperforms on all subjects. Assume that no two of them have the same score on the same subject, find the probability that there exists a pair of domination in class.
of positive integers that satisfy the equation![\[
x + xy + xyz = 31.
\]](http://latex.artofproblemsolving.com/a/7/8/a78d1e9d66831f11e7de08c109855e7727ec8535.png)
be real numbers such that 
, and![\[
a + \frac{1}{a} + b + \frac{1}{b} + c + \frac{1}{c} + d + \frac{1}{d} = 0.
\]](http://latex.artofproblemsolving.com/a/9/0/a90c94d82e3650560635616dacaf392d300b8c91.png)
Prove that one of the numbers 
or 
is equal to 
.
be an integer. Show that, if 
is an integer, then it is a perfect square.[/quote]
. If is an integer, then 
is at least rational, so that 
must be a perfect square then. Using Conway's topograph method, I have found out that the smallest non-negative pairs 
for which this happens are 
and 
, and that, for every such pair 
, the "next" such pair can be calculated as
The eigenvalues of that matrix are irrational, however, so that any calculation which uses powers of that matrix is a little cumbersome. There must be an easier way, but I cannot find it. Can you?
numbers placed in a circular way is either 
or 
. (Both values appear). Consider the total sum of the products of every 
consecutive numbers.
Find the minimal possible value of this sum.
Find the maximal possible value of this sum.
the orthocenter of . Construct line 
through perpendicular to 
. Construct the line through perpendicular to 
and let it intersect at 
. Let the line though perpendicular to meet 
respectively at 
. These are the required points. the orthocenter of 
and using that we finally constructed 
,the miquel point of 
is the infinity point of the newton line so we can construct 
then we draw a line parallel to through 
; it hits 
at another point say 
let the line 
cuts 
at point that is exactly . Finally we draw a parallel to to construct 
.
and two vectors 
, construct a line , with direction 
, such that Newton line of the quadrilateral formed by the four lines 
has direction 
. form a triangle . Let 
be the parabola inscribed in its medial triangle and whose axis has the direction of the vector 
. Let be the conjugate diameter of the direction of the vector . and intersects at a point 
. The tangent at to is Newton's line of the quadrilateral 
, which has the direction of the vector .
,
which satisfies
be two lines .
s.t. 
and the newton line of 
is parallel to 