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Topic
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real+ FE
pomodor_ap 4
N
by jasperE3
Source: Own, PDC001-P7
Let 
be a function such that
for all 
. Determine all such functions 
.
be a function such that
for all 
. Determine all such functions 
.
4 replies
FE solution too simple?
Yiyj1 8
N
by lksb
Source: 101 Algebra Problems from the AMSP
Find all functions 
such that the equality 
holds for all pairs of real numbers 
.
I feel like my solution is too simple. Is there something I did wrong or something I missed?
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.I feel like my solution is too simple. Is there something I did wrong or something I missed?
8 replies
Polynomials in Z[x]
BartSimpsons 16
N
by bin_sherlo
Source: European Mathematical Cup 2017 Problem 4
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.
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.
16 replies
Why is the old one deleted?
EeEeRUT 13
N
by EVKV
Source: EGMO 2025 P1
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
, 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
13 replies
Factor sums of integers
Aopamy 2
N
by cadaeibf
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.
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.
2 replies
Least integer T_m such that m divides gauss sum
Al3jandro0000 33
N
by NerdyNashville
Source: 2020 Iberoamerican P2
Let 
denotes the least natural such that
Find all naturals 
such that 
.
Proposed by Nicolás De la Hoz
denotes the least natural such that
Find all naturals 
such that 
.Proposed by Nicolás De la Hoz
33 replies
Estonian Math Competitions 2005/2006
STARS 2
N
by jasperE3
Source: Juniors Problem 4
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?
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?
2 replies
Sum of whose elements is divisible by p
nntrkien 43
N
by lpieleanu
Source: IMO 1995, Problem 6, Day 2, IMO Shortlist 1995, N6
Let 
be an odd prime number. How many -element subsets 
of 
are there, the sum of whose elements is divisible by ?
be an odd prime number. How many -element subsets 
of 
are there, the sum of whose elements is divisible by ?
43 replies
Arrangement of integers in a row with gcd
egxa 2
N
by Qing-Cloud
Source: All Russian 2025 10.5 and 11.5
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?
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?2 replies
Integer representation
RL_parkgong_0106 1
N
by Jackson0423
Source: Own
Show that for any positive integer , there exists some positive integer that makes the following equation have no integer root 
.
, there exists some positive integer that makes the following equation have no integer root 
.
1 reply
