satisfy that 
.
Function on positive integers with two inputs
by Assassino9931, Jan 27, 2025, 10:03 AM
The function 
is such that 
for any positive integers 
. Assume there exists a positive integer 
such that 
for all positive integers 
. Determine all possible values of 
.
is such that 
for any positive integers 
. Assume there exists a positive integer 
such that 
for all positive integers 
. Determine all possible values of 
.This post has been edited 1 time. Last edited by Assassino9931, Jan 27, 2025, 10:06 AM
Concurrency
by Dadgarnia, Mar 12, 2020, 10:54 AM
Let 
be an isosceles triangle (
) with incenter 
. Circle 
passes through 
and and is tangent to 
. intersects 
and circumcircle of at 
and 
, respectively. Let 
be the midpoint of 
and 
be the midpoint of 
. Prove that 
, 
and 
are concurrent.
Proposed by Alireza Dadgarnia
be an isosceles triangle (
) with incenter 
. Circle 
passes through 
and and is tangent to 
. intersects 
and circumcircle of at 
and 
, respectively. Let 
be the midpoint of 
and 
be the midpoint of 
. Prove that 
, 
and 
are concurrent.Proposed by Alireza Dadgarnia
Nice inequality
by sqing, Apr 24, 2019, 1:01 PM
Let 
be real numbers . Prove that : There exist positive integer 
such that 
Where 
be real numbers . Prove that : There exist positive integer 
such that 
Where 
Tiling rectangle with smaller rectangles.
by MarkBcc168, Jul 10, 2018, 11:25 AM
A rectangle 
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.
Proposed by Jeck Lim, Singapore
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.Proposed by Jeck Lim, Singapore
This post has been edited 2 times. Last edited by MarkBcc168, Jul 15, 2018, 12:57 PM
Two circles, a tangent line and a parallel
by Valentin Vornicu, Oct 24, 2005, 10:15 AM
Two circles 
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
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