My problem that I could not find(NT)
by Nuran2010, Apr 24, 2025, 1:49 PM
While I was thinking on some other geometry problem, a NT problem came to my mind. Despite some tries(which were mostly order), I could not find a way to solve the problem. As I searched, this problem has never been posted before. Here is the problem.
Find all positive integers
such that:
Moreover, I wonder if there is a way to solve the question in this variant:
Find all positive integers
such that:
Find all positive integers
such that:
Moreover, I wonder if there is a way to solve the question in this variant:
Find all positive integers
such that:
is given, 
,
are graphs on the same set of 
nodes, having 
connected components respectively. Prove that 
has at least 
connected components.
,
,
are pairwise distinct real numbers satisfying 
Prove that
Let 
Prove that
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
and 
be positive integers.
and 
.
.
and 
intersect at points 
and 
,
intersects 
,
intersects 
.
is the circumcenter of 
,
.
,
units horizontally (left or right); or
to 
using at most 
moves.
and a subset of the numbers written on the board and adds 
be real numbers . Prove that : There exist positive integer 
such that 
Where 
first, then every player sees the last number written and if it is 
or 
,
.
wins for 
and 
.
in expression of 
.
be pairwise coprime positive integers with 
being prime and 
.![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by 
September 2017
February 2017