G
Topic
First Poster
Last Poster
Linear recurrence fits with factorial finitely often
Assassino9931 2
N
by Assassino9931
Source: Bulgaria Balkan MO TST 2025
Let 
be an integer. The sequence 
is defined via 
, 
and
![\[ a_{n+2} = ka_{n+1} + a_n \]](//latex.artofproblemsolving.com/b/5/e/b5e3e6f9f0b0709113498266e41b372299a2986f.png)
for any positive integer 
. Prove that there are finitely many pairs 
of positive integers such that 
.
be an integer. The sequence 
is defined via 
, 
and![\[ a_{n+2} = ka_{n+1} + a_n \]](http://latex.artofproblemsolving.com/b/5/e/b5e3e6f9f0b0709113498266e41b372299a2986f.png)
for any positive integer 
. Prove that there are finitely many pairs 
of positive integers such that 
.
2 replies
Rational Number Theory
Olympiadium 11
N
by whwlqkd
Source: KJMO 2022 P8
Find all pairs 
of rational numbers such that 
of rational numbers such that 
11 replies
Stereotypical Diophantine Equation
Mathdreams 2
N
by grupyorum
Source: 2025 Nepal Mock TST Day 2 Problem 1
Find all solutions in the nonnegative integers to 
.
(Shining Sun, USA)
.(Shining Sun, USA)
2 replies
An easy 3 variable equation
BarisKoyuncu 6
N
by Burak0609
Source: Turkey National Mathematical Olympiad 2022 P4
For which real numbers 
, there exist pairwise different real numbers 
satisfying
, there exist pairwise different real numbers 
satisfying
6 replies
Junior Balkan Mathematical Olympiad 2024- P3
Lukaluce 13
N
by EVKV
Source: JBMO 2024
Find all triples of positive integers 
that satisfy the equation
Proposed by Ognjen Tešić, Serbia
that satisfy the equation
Proposed by Ognjen Tešić, Serbia
13 replies
solve in Z: xy = 3 (x+y)-1
parmenides51 5
N
by fruitmonster97
Source: Greece JBMO TST 2008 p4
Product of two integers is 
less than three times of their sum. Find those integers.
less than three times of their sum. Find those integers.
5 replies
Israel Number Theory
mathisreaI 62
N
by cursed_tangent1434
Source: IMO 2022 Problem 5
Find all triples 
of positive integers with 
prime and ![\[ a^p=b!+p. \]](//latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
of positive integers with 
prime and ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
62 replies
Hard number theory
Hip1zzzil 13
N
by Hip1zzzil
Source: FKMO 2025 P6
Positive integers 
satisfy both of the following conditions.
For a positive integer
, if 
, then 
.
There exist integers
that satisfies the equation 
and 
.
Prove that there exist integers that satisfies the equation 
, for each integer .
satisfy both of the following conditions.For a positive integer
, if 
, then 
.There exist integers
that satisfies the equation 
and 
.Prove that there exist integers
that satisfies the equation 
, for each integer .
13 replies
weird looking system of equations
Valentin Vornicu 37
N
by deduck
Source: USAMO 2005, problem 2, Razvan Gelca
Prove that the system 
has no solutions in integers 
, 
, and 
.
has no solutions in integers 
, 
, and 
.
37 replies
k Factorials, Sum of Powers, Diophantine Equations,
John_Mgr 3
N
by John_Mgr
Source: Nepal NMO 2025 p4
Find all the pairs of positive integers and such that: ![\[1^n+2^n+3^n+\cdots +n^n=x!\]](//latex.artofproblemsolving.com/1/f/e/1fe2d44fea6376291c30257973e2e90deab76557.png)
and such that: ![\[1^n+2^n+3^n+\cdots +n^n=x!\]](http://latex.artofproblemsolving.com/1/f/e/1fe2d44fea6376291c30257973e2e90deab76557.png)
3 replies
