be circumcenter of a non-isosceles triangle 
be a point in the interior of 
.
be foots of perpendicular lines from 
.
is cyclic and 
is the circumcenter of 
.
bisects 
be a triangle with 
.
be the midpoint of 
,
be a point on segment 
.
be the point of intersection, different from 
,
and line 
.
and 
as the points of intersection of line 
with 
and 
,
.
with 
on side 
.
be the reflection of 
be a point on line 
such that lines 
and 
and 
meet again at 
.
.
such that 
holds for all real numbers 
and 
.
,
which satisfies
students and their test results on 
different subjects, we say that student 
if and only if 
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 
,![\[
a + \frac{1}{a} + b + \frac{1}{b} + c + \frac{1}{c} + d + \frac{1}{d} = 0.
\]](http://latex.artofproblemsolving.com/a/9/0/a90c94d82e3650560635616dacaf392d300b8c91.png)
Prove that one of the numbers 
or 
is equal to 
.
is an integer, then it is a perfect square.[/quote]
.
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 
,
,
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 
.
consecutive numbers.
Find the minimal possible value of this sum.
Find the maximal possible value of this sum.
people who do not know each other. Prove that it is possible to introduce them in such a way that no three of them have the same number of acquaintances.
,
and 
and 
respectively. The corresponding excircle touches the side 
.
be the common point of 
and the incircle, closest to 
be the common point of 
and 
.
.
be the 
-
and 
for 
)
,
be the smallest integer 
.
Y by Adventure10, Mango247
For a nonisosceles triangle 
, consider the altitude from vertex 
and two bisectrices from remaining vertices. Prove that the circumcircle of the triangle formed by these three lines touches the bisectrix from vertex .
, consider the altitude from vertex 
and two bisectrices from remaining vertices. Prove that the circumcircle of the triangle formed by these three lines touches the bisectrix from vertex .This post has been edited 1 time. Last edited by parmenides51, Apr 30, 2019, 7:19 PM
Reason: official formulation added
Reason: official formulation added
is point of intersection of bisectors and lies in the circumcircle, so we need to show, that 
so 
