be a triangle, and let 
and 
be points on segment 
symmetric with respect to the midpoint of 
denote the circle passing through 
at 
denote the circle passing through 
at 
)
.
the feet of the altitudes lying on 
respectively. One of the intersection points of the line 
and the circumcircle is 
The lines 
and 
meet at point 
Prove that 
and circumcenter 
.
and 
be points on the line segments 
is a parallelogram. Prove that 
.
cities, 
airline companies in a country. Between any two cities, there is exactly one 
and circumscribed about the incircle 
,
.
,
,
,
,
,
intersects 
(distinct from 
of 
at 
at 
.
.
but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer 
(not necessarily in 
.
chosen by Arnaldo.
be a set of 2025 positive real numbers. For a subset 
,
as the median of 
when all elements of 
.![\[
P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.
\]](http://latex.artofproblemsolving.com/1/d/f/1df2ea8487289111939a53dcc49576fbe6fdef37.png)
Find the smallest real number 
such that for any set ![\[
P(A) - Q(A) \leq C \cdot \max(A),
\]](http://latex.artofproblemsolving.com/c/d/e/cdea343d11c722a423e857aeb01202863405773e.png)
where 
denotes the largest element in 
in 
colors, there exists an arithmetic progression of size 
.
from this?
be foots of perpendicular lines from 
.
is cyclic and 
.
bisects 
such that 
interesting if 
of 
and 
is also a divisor of 
Y by GoJensenOrGoHome, Adventure10
Hello everyone.
This is my first actual AOPS post, even though I'm a long time IMO competitor and an avid AOPS user.
If you don't want to read the Problem History, you can scroll down to read the statement right away.
Problem History
The reason why I'm writing this post is to publish a very interesting geometry problem, which I've created with the help of a group of friends of mine. I started creating the problem around 2 years ago, when I heard about another geometry problem, and I developed the idea used in that problem into something much more, together with two of my friends. We reached some interesting conclusions, but couldn't prove any of it, so we stopped working on it after a month or so. I continued working on that problem alone for another two months maybe, and then I let it go.
About half a year ago, while I was on a math competition abroad, I told one of my teammates about the problem and he was interested in solving it together with me, so we resumed the investigation. He proposed one crucial idea that reshaped the problem into something totally different, and after I'd noticed one more important fact, we completely formed the problem. We still couldn't prove any of it, because it all seemed too difficult. After the competition, we stopped working on it.
Finally, around two weeks ago, I remembered the problem, and used Wolfram Mathematica to prove it. And guess what... Our hunch was indeed correct. A brand new geometry problem was born, and I was quite satisfied. That means that our solution to the problem was indeed correct, since we finally got a computerized proof. But to me, it's still not over. I want to know if there are any normal ways to prove this...Without using extreme analytic computations performed by a sophisticated mathematical program.
That is why I'm posting this problem as a challenge to everyone else who wants to try and solve it. Maybe someone here can pull it off? I don't know. I've shown this problem to a lot of successful IMO competitors, and none of them were close to solving it. I honestly think the problem is difficult, but I don't know how difficult it actually is. Anyhow, I'm going to post the problem statement text now. Good luck and best wishes to everyone!
Problem Statement
Three fixed points,
, 
and 
, are given such that 
is a scalene triangle. A variable real parameter 
is also given, and six points, 
, 
, 
, 
, 
and 
, are defined by the following six vector equalities:
Let
be the circumcenter of triangle 
, and 
the circumcenter of triangle 
. The point 
is defined as the midpoint of 
.
For any fixed triple of points, and , find the locus of , as takes all possible real values.
This is my first actual AOPS post, even though I'm a long time IMO competitor and an avid AOPS user.
If you don't want to read the Problem History, you can scroll down to read the statement right away.
Problem History
The reason why I'm writing this post is to publish a very interesting geometry problem, which I've created with the help of a group of friends of mine. I started creating the problem around 2 years ago, when I heard about another geometry problem, and I developed the idea used in that problem into something much more, together with two of my friends. We reached some interesting conclusions, but couldn't prove any of it, so we stopped working on it after a month or so. I continued working on that problem alone for another two months maybe, and then I let it go.
About half a year ago, while I was on a math competition abroad, I told one of my teammates about the problem and he was interested in solving it together with me, so we resumed the investigation. He proposed one crucial idea that reshaped the problem into something totally different, and after I'd noticed one more important fact, we completely formed the problem. We still couldn't prove any of it, because it all seemed too difficult. After the competition, we stopped working on it.
Finally, around two weeks ago, I remembered the problem, and used Wolfram Mathematica to prove it. And guess what... Our hunch was indeed correct. A brand new geometry problem was born, and I was quite satisfied. That means that our solution to the problem was indeed correct, since we finally got a computerized proof. But to me, it's still not over. I want to know if there are any normal ways to prove this...Without using extreme analytic computations performed by a sophisticated mathematical program.
That is why I'm posting this problem as a challenge to everyone else who wants to try and solve it. Maybe someone here can pull it off? I don't know. I've shown this problem to a lot of successful IMO competitors, and none of them were close to solving it. I honestly think the problem is difficult, but I don't know how difficult it actually is. Anyhow, I'm going to post the problem statement text now. Good luck and best wishes to everyone!
Problem Statement
Three fixed points,
, 
and 
, are given such that 
is a scalene triangle. A variable real parameter 
is also given, and six points, 
, 
, 
, 
, 
and 
, are defined by the following six vector equalities:
Let
be the circumcenter of triangle 
, and 
the circumcenter of triangle 
. The point 
is defined as the midpoint of 
.For any fixed triple of points
, and , find the locus of , as takes all possible real values.This post has been edited 1 time. Last edited by IvanGD, Nov 4, 2015, 6:42 PM
.
on 
,
and 
.
