Integer representation
by RL_parkgong_0106, Apr 22, 2025, 2:23 PM
Show that for any positive integer
, there exists some positive integer
that makes the following equation have no integer root
.





Arrangement of integers in a row with gcd
by egxa, Apr 18, 2025, 5:09 PM
Let
be a natural number. The numbers
are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the
GCDs obtained?



This post has been edited 1 time. Last edited by egxa, Apr 18, 2025, 5:18 PM
Why is the old one deleted?
by EeEeRUT, Apr 16, 2025, 1:33 AM
For a positive integer
, let
be all positive integers smaller than
that are coprime to
. Find all
such that
for all 
Here
is the largest positive integer that divides both
and
. Integers
and
are coprime if
.
Proposed by Paulius Aleknavičius, Lithuania







Here






Proposed by Paulius Aleknavičius, Lithuania
This post has been edited 2 times. Last edited by EeEeRUT, Apr 18, 2025, 12:56 AM
Reason: Authorship
Reason: Authorship
Factor sums of integers
by Aopamy, Feb 23, 2023, 3:13 AM
Let
be a positive integer. A positive integer
is called a benefactor of
if the positive divisors of
can be partitioned into two sets
and
such that
is equal to the sum of elements in
minus the sum of the elements in
. Note that
or
could be empty, and that the sum of the elements of the empty set is
.
For example,
is a benefactor of
because
.
Show that every positive integer
has at least
benefactors.












For example,



Show that every positive integer


Least integer T_m such that m divides gauss sum
by Al3jandro0000, Nov 17, 2020, 7:24 PM
Let
denotes the least natural such that
Find all naturals
such that
.
Proposed by Nicolás De la Hoz




Proposed by Nicolás De la Hoz
This post has been edited 3 times. Last edited by Al3jandro0000, Nov 18, 2020, 3:54 PM
Reason: Some clarifications
Reason: Some clarifications
Polynomials in Z[x]
by BartSimpsons, Dec 27, 2017, 12:25 PM
Find all polynomials
with integer coefficients such that
and
is a square of an integer for all nonnegative integers
.
Remark: For a nonnegative integer
and an integer
,
is defined as follows:
if
and
if
.
Proposed by Adrian Beker.




Remark: For a nonnegative integer







Proposed by Adrian Beker.
This post has been edited 1 time. Last edited by BartSimpsons, Dec 27, 2017, 12:26 PM
Reason: added source
Reason: added source
Factor of P(x)
by Brut3Forc3, Apr 4, 2010, 2:45 AM
If
, and
are all polynomials such that
prove that
is a factor of
.


![\[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\]](http://latex.artofproblemsolving.com/1/6/a/16a71abc110ece558d427a07d7be2d1f72148024.png)


Estonian Math Competitions 2005/2006
by STARS, Jul 30, 2008, 1:17 AM
A
square is divided into unit squares. Is it possible to fill each unit square with a number
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?


Sum of whose elements is divisible by p
by nntrkien, Aug 8, 2004, 1:29 AM
Let
be an odd prime number. How many
-element subsets
of
are there, the sum of whose elements is divisible by
?





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