be an acute triangle. Define 
as its incenter. Let 
and 
be the incircle's tangent points to 
and 
,
be the midpoint of 
.
be the intersection point of a perpendicular line passing through 
.
intersects the circumcircle of 
at 
.
intersects line 
at 
.
is cyclic.
for which there exists a sequence of distinct positive integers 
,
,
,
satisfying ![\[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]](http://latex.artofproblemsolving.com/a/f/5/af594ff07aa2f89cb70ae669263558049556cf0e.png)
be an integer and 
is a sequence of real numbers for which 
and:
Prove that all elements of the sequence are positive.
be a positive integer, and let 
.
parts is a representation of n as a sum 
,
belong to 
.
parts with 
.![$\sqrt[3]{6n}$](http://latex.artofproblemsolving.com/2/0/1/201489b1f95ad200036698a606f712da7bb465b9.png)
different parts.
be 
,
?
through 
,
such that 
is also prime.
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
such that 
.
,
.
and 
such that the only 
-
equals its 
-
.
(with 
)
,
,
.
be the midpoint of 
.
at 
.
,
in degrees?
with 
and 
for 
.
.
.
.
(x) be the 
.
such that 
divides 
?
be the set of vectors 
where 
are all real numbers. Let 
denote 
.
be the set in 
given by 
If a point 
is uniformly at random from ![$E[||z||^2]$](http://latex.artofproblemsolving.com/e/1/a/e1aa966627c05d1f317981d82094e07d1b1ce0e1.png)
?
be the unique integer between 
and 
,
.
and 
,
.
,
,
,
.
,
and the puck bounces perfectly around the box (angle in equals angle out, no friction) until it hits one of the four vertices of the box. What is the expected number of times it will hit an edge or vertex of the box, including both when it starts at 
.
,
,
.
graph paper notebooks. How many ways can she buy school supplies?
.
be the midpoint of minor arc 
and let 
be a point on line 
.
and the Circle 
and 
.
and 
,
.
to be the maximum possible least-common-multiple of any sequence of positive integers which sum to 
but I am looking for the exact value. Does anyone know how to solve this problem?
continuous function with 
for all real 
.
for all real 
,
?
s.t. 
and 
be positiv integer numbers.
Y by
Given is an acute and scalene triangle 
with circumcenter 
. 
and 
intersect the altitude from 
to 
at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
with circumcenter 
. 
and 
intersect the altitude from 
to 
at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.This post has been edited 1 time. Last edited by Mahdi_Mashayekhi, Apr 19, 2025, 11:25 AM
concyclic .
concyclic.
as the second intersection of 
and 
.
.
hence 
is concyclic. Similarly, 
is concyclic, and we are done.
and 
so 
.
so 
is cyclic and since 
then 
are collinear as wanted.
