by EmersonSoriano, Apr 2, 2025, 10:08 PM 
given that there exist positive integers 
satisfying
by EmersonSoriano, Apr 2, 2025, 10:03 PM 
and 
, we define 
as follows:
where 
are all the positive divisors of . For example,
.
Find all positive integers such that 
.
Find all positive integers such that 
.by EmersonSoriano, Apr 2, 2025, 9:58 PM 
be a fixed square and 
a variable point on segment 
. The square 
is constructed such that 
is on segment 
and 
is on segment 
. Let 
be the intersection point of lines 
and 
. Find the locus of as varies along segment .by EmersonSoriano, Apr 2, 2025, 9:56 PM be a positive integer. In an 
board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place queens on the board such that no two threaten each other when: 
. 
.by Ciobi_, Apr 2, 2025, 12:56 PM 
be a point in the plane, distinct from the vertices of 
. Consider 
the reflections of with respect to lines 
and 
, in this order.
are collinear if and only if lies on the circumcircle of . does not lie on the circumcircle of and the centroids of triangles and 
coincide, prove that is equilateral.by dotscom26, Apr 1, 2025, 4:11 AM 
be real numbers satisfying
by Ruji2018252, Mar 26, 2025, 8:45 AM 
and![\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/c/8/9/c895ae7fdf8d7f284ac9fc94cc077d6edad6cbf0.png)
by AndreiVila, Mar 8, 2025, 1:28 PM 
be a convex hexagon with 
and 
.
which is equidistant from sides 
and 
.
and 
are the centers of mass of 
and 
, show that 
.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 ?"Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu
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