Y by
Let 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](//latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
G
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First Poster
Last Poster
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
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
Divisibility..
Sadigly 4
N
by Solar Plexsus
Source: another version of azerbaijan nmo 2025
Just ignore this
4 replies
Surjective number theoretic functional equation
snap7822 3
N
by internationalnick123456
Source: 2025 Taiwan TST Round 3 Independent Study 2-N
Let 
be a function satisfying the following conditions:
[list=i]
[*] For all
, if 
and 
, then 
;
[*]
is surjective.
[/list]
Find the maximum possible value of
.
Proposed by snap7822
be a function satisfying the following conditions:[list=i]
[*] For all
, if 
and 
, then 
;[*]
is surjective.[/list]
Find the maximum possible value of
.Proposed by snap7822
3 replies
Many Equal Sides
mathisreal 3
N
by QueenArwen
Source: Brazil EGMO TST 2023 #1
Let be a triangle with 
and 
. Let and be the midpoints of 
and 
respectively. The point is inside of 
such that 
is equilateral. Let 
and 
. Prove that 
.
be a triangle with 
and 
. Let and be the midpoints of 
and 
respectively. The point is inside of 
such that 
is equilateral. Let 
and 
. Prove that 
.
3 replies
LOTS of recurrence!
SatisfiedMagma 4
N
by Reacheddreams
Source: Indian Statistical Institute Entrance UGB 2023/5
There is a rectangular plot of size 
. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size 
, the blue tiles are of size and the black tiles are of size 
. Let 
denote the number of ways this can be done. For example, clearly 
because we can have either a red or a blue tile. Also 
since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
[list=a]
[*]Prove that
for all 
.
[*]Prove that
for all 
.
[/list]
Here,
![\[ \binom{m}{r} = \begin{cases}
\dfrac{m!}{r!(m-r)!}, &\text{ if $0 \le r \le m$,} \\
0, &\text{ otherwise}
\end{cases}\]](//latex.artofproblemsolving.com/b/8/d/b8d653661e421db1fce7d93837e7bce31337bb30.png)
for integers 
.
. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size 
, the blue tiles are of size and the black tiles are of size 
. Let 
denote the number of ways this can be done. For example, clearly 
because we can have either a red or a blue tile. Also 
since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.[list=a]
[*]Prove that
for all 
.[*]Prove that
for all 
.[/list]
Here,
![\[ \binom{m}{r} = \begin{cases}
\dfrac{m!}{r!(m-r)!}, &\text{ if $0 \le r \le m$,} \\
0, &\text{ otherwise}
\end{cases}\]](http://latex.artofproblemsolving.com/b/8/d/b8d653661e421db1fce7d93837e7bce31337bb30.png)
for integers 
.
4 replies
Inequality with three conditions
G
H
G
H
Source: Romania EGMO TST 2019 Day 1 P3
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completes it.
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Y by
oVlad wrote:
Let be non-negative real numbers such that Prove that
be non-negative real numbers such that Prove that https://artofproblemsolving.com/community/c6h1754566p11450296
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