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Non-homogenous Inequality
Adywastaken 7
N
by ehuseyinyigit
Source: NMTC 2024/7
such that 
. Show that 
. When will equality hold?
7 replies
FE with devisibility
fadhool 2
N
by ATM_
if when i solve an fe that is defined in the set of positive integer i found m|f(m) can i set f(m) =km such that k is not constant and of course it depends on m but after some work i find k=c st c is constant is this correct
2 replies
Iranian geometry configuration
Assassino9931 2
N
by Captainscrubz
Source: Al-Khwarizmi Junior International Olympiad 2025 P7
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
2 replies
f(m + n) >= f(m) + f(f(n)) - 1
orl 30
N
by ezpotd
Source: IMO Shortlist 2007, A2, AIMO 2008, TST 2, P1, Ukrainian TST 2008 Problem 8
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
30 replies
Classic Diophantine
Adywastaken 3
N
by Adywastaken
Source: NMTC 2024/6
Find all natural number solutions to 
.
.
3 replies
Add d or Divide by a
MarkBcc168 25
N
by Entei
Source: ISL 2022 N3
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 
.
25 replies
Alice and Bob play, 8x8 table, white red black, minimum n for victory
parmenides51 14
N
by Ilikeminecraft
Source: JBMO Shortlist 2018 C3
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.14 replies
GEOMETRY GEOMETRY GEOMETRY
Kagebaka 71
N
by bin_sherlo
Source: IMO 2021/3
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.
71 replies
Equation of integers
jgnr 3
N
by KTYC
Source: Indonesia Mathematics Olympiad 2005 Day 1 Problem 2
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 
.
3 replies
