[/quote]
,
,
,
be the midpoints of sides BC, CA, and AB, respectively. Point 
is located on the ray 
so that 
is similar to 
.
and 
on rays 
and 
,
,
,
are concurrent, i.e. these three lines intersect at a point.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
be a triangle. Distinct points 
lie on sides 
and 
are cyclic. Line 
meets the circumcircle of 
again at 
.
such that 
,
with 
is similar to 
.
.
be the set of real numbers. Find all functions 
that satisfy the following condition. Here, 
is the function obtained by composing 
times, that is, 
,
be an odd prime number and 
be the set of positive integers. Suppose that a function 
satisfies the following properties:
.
for any pair of relatively prime positive integers 
not both equal to 1;
for any pair of relatively prime positive integers 
.
be a point on 
such that 
.
the point on 
.
is the projection of 
passes through the orthocenter of 
.
and 
,
be the point on line 
such that the line through 
be the point on line 
such that the line through 
intersect the circumcircle of triangle 
.
and 
.
,
has orthocenter 
,
,
.
has orthocenter 
is a pair of real numbers such that
.
be a positive integer. Initially, a bishop is placed in each square of the top row of a 
to 
from left to right. A jump is a simultaneous move made by all bishops such that each bishop moves diagonally, in a straight line, some number of squares, and at the end of the jump, the bishops all stand in different squares of the same row.
of the numbers 
with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order 
,
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
Y by Rounak_iitr
Let 
be the incircle of 
. There are two smaller circles 
amd 
inside . The circle is tangent to at 
, tangent to 
at 
, and also tangent to 
. The circle is tangent to at 
, tangent to at 
, and also tangent to 
. Prove that 
are concyclic.
be the incircle of 
. There are two smaller circles 
amd 
inside . The circle is tangent to at 
, tangent to 
at 
, and also tangent to 
. The circle is tangent to at 
, tangent to at 
, and also tangent to 
. Prove that 
are concyclic.This post has been edited 1 time. Last edited by parmenides51, Jul 21, 2024, 7:46 AM
Reason: title typo
Reason: title typo
invert to two circles between the parallel lines.
is a diameter of the incirle
with centre 
is the tangency point of 
so 
are collinear. Similarly, 
are collinear
are concyclic. We are done.
![[asy]
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[/asy]](http://latex.artofproblemsolving.com/f/9/8/f9824f68fe9f0d8f0383e7475e8e47a9e2e759b0.png)
,
as the intersection of the tangent to 
at 
with 
,
at 
,
.
is the reflection of 
over 
.
have circumcenter 
,
,
be defined as the reflection of 
is cyclic.![\[\angle PDE = \angle PDH = 90 - \frac{\angle PHD}{2} = \frac{\angle PHZ}{2} = 90 - \angle PZH = 90 - \angle PQZ\]](http://latex.artofproblemsolving.com/2/e/a/2ead8032cac106bb13a69712430113415937ec4e.png)
However since 
is a diameter of 
we know that 
,![\[\angle PQE = 90 + \angle PQZ \Longrightarrow 180 - \angle PQE = 90 - \angle PQZ = \angle PDE\]](http://latex.artofproblemsolving.com/0/2/4/024384da3ab7afc9bd277d66ccd7369c20280366.png)
Thus 
