by pomodor_ap, Apr 21, 2025, 11:24 AM 
be a function such that
for all 
. Determine all such functions 
.by egxa, Apr 18, 2025, 5:09 PM 
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?by egxa, Apr 18, 2025, 9:58 AM 
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)by EeEeRUT, Apr 16, 2025, 1:33 AM 
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.by Yiyj1, Apr 9, 2025, 3:26 AM 
such that the equality 
holds for all pairs of real numbers 
.by Aopamy, Feb 23, 2023, 3:13 AM 
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 
.
is a benefactor of 
because 
. has at least 
benefactors.by Al3jandro0000, Nov 17, 2020, 7:24 PM 
denotes the least natural such that
Find all naturals 
such that 
.by BartSimpsons, Dec 27, 2017, 12:25 PM 
with integer coefficients such that 
and 
is a square of an integer for all nonnegative integers 
. and an integer , 
is defined as follows: 
if 
and 
if 
.by STARS, Jul 30, 2008, 1:17 AM 
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?by nntrkien, Aug 8, 2004, 1:29 AM 
be an odd prime number. How many -element subsets 
of 
are there, the sum of whose elements is divisible by ?A blog documenting a (no longer) high school youth and his struggles with advancing his mathematical skill.
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is an obvious solution. Now, let 
.
or 
.
.