Number Theory
by TUAN2k8, Apr 18, 2025, 3:25 PM
Find all positve integers m such that 
Inspired by 1984 IMO Problem 1
by sqing, Apr 18, 2025, 3:22 PM
Let 
and 
Prove that
and 
Prove that
This post has been edited 1 time. Last edited by sqing, 15 minutes ago
Inspired by 1984 IMO Problem 1
by sqing, Apr 18, 2025, 3:12 PM
Let and 
Prove that
and 
Prove that
This post has been edited 1 time. Last edited by sqing, 8 minutes ago
Prove that
Sum of divisors
by DinDean, Apr 18, 2025, 2:47 PM
Does there exist 
, such that 
, there exists an integer 
satisfying 
?
the sum of all positive divisors of .
, such that 
, there exists an integer 
satisfying 
?
the sum of all positive divisors of .Existence of perfect squares
by egxa, Apr 18, 2025, 9:48 AM
Find all natural numbers 
for which there exists an even natural number 
such that the number
![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](//latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.
for which there exists an even natural number 
such that the number![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](http://latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.3D russian combo
by egxa, Apr 18, 2025, 9:41 AM
A natural number 
is given. A cube with side length 
is made up of 
unit cubes, each of which is either black or white. It turns out that among any 
cubes that share a common vertex and form a 
cube, there are at most 
black cubes. What is the maximum number of black cubes that could have been used?
is given. A cube with side length 
is made up of 
unit cubes, each of which is either black or white. It turns out that among any 
cubes that share a common vertex and form a 
cube, there are at most 
black cubes. What is the maximum number of black cubes that could have been used?A problem with non-negative a,b,c
by KhuongTrang, Mar 4, 2025, 3:50 PM
Problem. Let 
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
Circles tangent to AD and AB intersect on AC
by gghx, Aug 3, 2024, 2:33 AM
In an acute triangle 
, 
, 
is the point on 
such that 
. Let 
be the circle through 
tangent to 
at , and 
the circle through tangent to 
at 
. Let 
be the second intersection of and . Prove that 
lies on 
.
, 
, 
is the point on 
such that 
. Let 
be the circle through 
tangent to 
at , and 
the circle through tangent to 
at 
. Let 
be the second intersection of and . Prove that 
lies on 
.Constructing sequences
by SMOJ, Mar 31, 2020, 7:09 AM
Starting with any -tuple 
, 
, of symbols from 
, we define a sequence 
according to the following rule: If 
, then 
, where 
if 
(taking 
) and 
is the symbol other than 
if 
. Find all positive integers 
for which there exists some integer 
such that 
.
-tuple 
, 
, of symbols from 
, we define a sequence 
according to the following rule: If 
, then 
, where 
if 
(taking 
) and 
is the symbol other than 
if 
. Find all positive integers 
for which there exists some integer 
such that 
.
October 2018
April 2018
April 2016
December 2014
October 2014
September 2014
August 2014