G
Topic
First Poster
Last Poster
A number theory problem
super1978 0
Source: Somewhere
Let 
be positive integers such that ![$\sqrt[n]{a}+\sqrt[n]{b}$](//latex.artofproblemsolving.com/d/a/a/daaf95cd3381060559529aa4a318e97b727eac31.png)
is an integer. Prove that 
are both the 
th power of 
positive integers.
be positive integers such that ![$\sqrt[n]{a}+\sqrt[n]{b}$](http://latex.artofproblemsolving.com/d/a/a/daaf95cd3381060559529aa4a318e97b727eac31.png)
is an integer. Prove that 
are both the 
th power of 
positive integers.
0 replies
1 viewing
A bit tricky invariant with 98 numbers on the board.
Nuran2010 3
N
by Nuran2010
Source: Azerbaijan Al-Khwarizmi IJMO TST 2025
The numbers 
are written on the board.In each step,two random numbers 
and 
are chosen and deleted.Then,the number 
is written instead.What will be the number remained on the board after the last step.
are written on the board.In each step,two random numbers 
and 
are chosen and deleted.Then,the number 
is written instead.What will be the number remained on the board after the last step.
3 replies
A irreducible polynomial
super1978 0
Source: Somewhere
Let 
such that 
is a prime number and 
. Prove that 
is irreducible over ![$\mathbb{Z}[x]$](//latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.
such that 
is a prime number and 
. Prove that 
is irreducible over ![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.
0 replies
(2^n + 1)/n^2 is an integer (IMO 1990 Problem 3)
orl 107
N
by Rayvhs
Source: IMO 1990, Day 1, Problem 3, IMO ShortList 1990, Problem 23 (ROM 5)
Determine all integers 
such that
![\[ \frac {2^n + 1}{n^2}
\]](//latex.artofproblemsolving.com/9/2/2/922444e7669c326efcad5fde3ea00f8c2d9fa5d8.png)
is an integer.
such that![\[ \frac {2^n + 1}{n^2}
\]](http://latex.artofproblemsolving.com/9/2/2/922444e7669c326efcad5fde3ea00f8c2d9fa5d8.png)
is an integer.
107 replies
n + k are composites for all nice numbers n, when n+1, 8n+1 both squares
parmenides51 2
N
by Assassino9931
Source: 2022 Saudi Arabia JBMO TST 1.1
The positive 
called ‘nice’ if and only if 
and 
are both perfect squares. How many positive integers 
such that 
are composites for all nice numbers ?
called ‘nice’ if and only if 
and 
are both perfect squares. How many positive integers 
such that 
are composites for all nice numbers ?
2 replies
Functional inequality condition
WakeUp 3
N
by AshAuktober
Source: Italy TST 1995
A function 
satisfies the conditions
![\[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\]](//latex.artofproblemsolving.com/5/5/6/556f545540d4941d058a1c4099b4b86eb35caf0d.png)
Prove that
for all real 
.
satisfies the conditions![\[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\]](http://latex.artofproblemsolving.com/5/5/6/556f545540d4941d058a1c4099b4b86eb35caf0d.png)
Prove that
for all real 
.
3 replies
Asymmetric FE
sman96 16
N
by jasperE3
Source: BdMO 2025 Higher Secondary P8
Find all functions 
such that
for all 
.
such that
for all 
.
16 replies
Existence of a rational arithmetic sequence
brianchung11 28
N
by cursed_tangent1434
Source: APMO 2009 Q.4
Prove that for any positive integer 
, there exists an arithmetic sequence 
of rational numbers, where 
are relatively prime positive integers for each 
such that the positive integers 
are all distinct.
, there exists an arithmetic sequence 
of rational numbers, where 
are relatively prime positive integers for each 
such that the positive integers 
are all distinct.
28 replies
NT from EGMO 2018
BarishNamazov 39
N
by cursed_tangent1434
Source: EGMO 2018 P2
Consider the set
![\[A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.\]](//latex.artofproblemsolving.com/3/d/f/3dfb1d52835627a2977f0074f8df7e7ed677f2a5.png)
[list=a]
[*]Prove that every integer
can be written as the product of one or more elements of 
, which are not necessarily different.
[*]For every integer let denote the minimum integer such that can be written as the
product of elements of , which are not necessarily different.
Prove that there exist infinitely many pairs
of integers with 
, 
, and ![\[f(xy)<f(x)+f(y).\]](//latex.artofproblemsolving.com/c/f/f/cff5554b8e72d99f8989448c2621794242d784d8.png)
(Pairs 
and 
are different if 
or 
).
[/list]
![\[A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.\]](http://latex.artofproblemsolving.com/3/d/f/3dfb1d52835627a2977f0074f8df7e7ed677f2a5.png)
[list=a]
[*]Prove that every integer
can be written as the product of one or more elements of 
, which are not necessarily different.[*]For every integer
let denote the minimum integer such that can be written as theproduct of
elements of , which are not necessarily different.Prove that there exist infinitely many pairs
of integers with 
, 
, and ![\[f(xy)<f(x)+f(y).\]](http://latex.artofproblemsolving.com/c/f/f/cff5554b8e72d99f8989448c2621794242d784d8.png)
(Pairs 
and 
are different if 
or 
).[/list]
39 replies
ISI UGB 2025 P6
SomeonecoolLovesMaths 2
N
by Mathgloggers
Source: ISI UGB 2025 P6
Let 
denote the set of natural numbers, and let 
, 
, be nine distinct tuples in 
. Show that there are three distinct elements in the set 
whose product is a perfect cube.
denote the set of natural numbers, and let 
, 
, be nine distinct tuples in 
. Show that there are three distinct elements in the set 
whose product is a perfect cube.
2 replies
