.Iranian geometry configuration
by Assassino9931, May 10, 2025, 9:39 AM
Let 
be a cyclic quadrilateral with circumcenter 
, such that 
is not a diameter of its circumcircle. The lines 
and 
intersect at point 
, so that 
lies between 
and , and 
lies between 
and . Suppose triangle 
is acute and let 
be its orthocenter. The points 
and 
on the lines and , respectively, are such that 
and 
. The line through , perpendicular to , intersects at 
, and the line through , perpendicular to , intersects at 
. Prove that the points , , are collinear.
Amir Parsa Hosseini Nayeri, Iran
be a cyclic quadrilateral with circumcenter 
, such that 
is not a diameter of its circumcircle. The lines 
and 
intersect at point 
, so that 
lies between 
and , and 
lies between 
and . Suppose triangle 
is acute and let 
be its orthocenter. The points 
and 
on the lines and , respectively, are such that 
and 
. The line through , perpendicular to , intersects at 
, and the line through , perpendicular to , intersects at 
. Prove that the points , , are collinear.Amir Parsa Hosseini Nayeri, Iran
This post has been edited 1 time. Last edited by Assassino9931, Today at 9:39 AM
Divisibility..
by Sadigly, May 9, 2025, 7:37 AM
This post has been edited 5 times. Last edited by Sadigly, 2 hours ago
Add d or Divide by a
by MarkBcc168, Jul 9, 2023, 4:40 AM
Let 
be a positive integer and 
be a positive integer coprime to 
. Let 
, and for 
, define
Find, in terms of and 
, the greatest positive integer 
for which there exists an index 
such that 
is divisible by 
.
be a positive integer and 
be a positive integer coprime to 
. Let 
, and for 
, define
Find, in terms of and 
, the greatest positive integer 
for which there exists an index 
such that 
is divisible by 
.GEOMETRY GEOMETRY GEOMETRY
by Kagebaka, Jul 20, 2021, 8:47 PM
Let be an interior point of the acute triangle 
with 
so that 
The point on the segment 
satisfies 
the point on the segment 
satisfies 
and the point 
on the line satisfies 
Let 
and 
be the circumcenters of the triangles 
and 
respectively. Prove that the lines 
and 
are concurrent.
be an interior point of the acute triangle 
with 
so that 
The point on the segment 
satisfies 
the point on the segment 
satisfies 
and the point 
on the line satisfies 
Let 
and 
be the circumcenters of the triangles 
and 
respectively. Prove that the lines 
and 
are concurrent.Alice and Bob play, 8x8 table, white red black, minimum n for victory
by parmenides51, Jul 22, 2019, 3:19 AM
The cells of a 
table are initially white. Alice and Bob play a game. First Alice paints of the fields in red. Then Bob chooses 
rows and columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of such that Alice can win the game no matter how Bob plays.
table are initially white. Alice and Bob play a game. First Alice paints of the fields in red. Then Bob chooses 
rows and columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of such that Alice can win the game no matter how Bob plays.f(m + n) >= f(m) + f(f(n)) - 1
by orl, Jul 13, 2008, 12:47 PM
Consider those functions 
which satisfy the condition
![\[ f(m + n) \geq f(m) + f(f(n)) - 1
\]](//latex.artofproblemsolving.com/4/b/d/4bddce124cf0da82cb8bb6c6ae2325239354f27c.png)
for all
Find all possible values of 
Author: Nikolai Nikolov, Bulgaria
which satisfy the condition![\[ f(m + n) \geq f(m) + f(f(n)) - 1
\]](http://latex.artofproblemsolving.com/4/b/d/4bddce124cf0da82cb8bb6c6ae2325239354f27c.png)
for all
Find all possible values of 
Author: Nikolai Nikolov, Bulgaria
This post has been edited 2 times. Last edited by orl, Feb 12, 2009, 5:18 PM
Equation of integers
by jgnr, Jun 2, 2008, 6:30 PM
For an arbitrary positive integer 
, define 
as the product of the digits of (in decimal). Find all positive integers such that 
.
, define 
as the product of the digits of (in decimal). Find all positive integers such that 
.
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