[/quote]
,
Choose a number of the form 
,
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say 
and erase the number in the other square.
be an acute-angled triangle with circumcircle 
and circumcentre 
.
and 
lie on 
and 
.
meet 
at 
,
meet 
at 
.
and 
have an intersection lie on line 
.
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)
be a convex hexagon with 
and 
.
which is equidistant from sides 
and 
.
and 
are the centers of mass of 
and 
,
.
be a point in the plane, distinct from the vertices of 
.
the reflections of 
and 
,
are collinear if and only if 
coincide, prove that 
given that there exist positive integers 
satisfying
and 
as follows:
where 
are all the positive divisors of 
.
Find all positive integers 
.
Find all positive integers 
.
be an odd prime number. How many 
of 
are there, the sum of whose elements is divisible by 
be real numbers satisfying
be a fixed square and 
a variable point on segment 
.
is constructed such that 
is on segment 
and 
is on segment 
.
be the intersection point of lines 
and 
.
board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place 
.
.
denote the number of positive integers less than 
has at least 
solutions in 
that can be written in the form
where 
are pairwise coprime positive integers such that 
,
,
.
Y by Phorphyrion, Zfn.nom-_nom
Let 
be a positive integer. Initially, a bishop is placed in each square of the top row of a 
chessboard; those bishops are numbered from
to 
from left to right. A jump is a simultaneous move made by all bishops such that each bishop moves diagonally, in a straight line, some number of squares, and at the end of the jump, the bishops all stand in different squares of the same row.
Find the total number of permutations
of the numbers 
with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order 
, from left to right.
Israel
be a positive integer. Initially, a bishop is placed in each square of the top row of a 
chessboard; those bishops are numbered from
to 
from left to right. A jump is a simultaneous move made by all bishops such that each bishop moves diagonally, in a straight line, some number of squares, and at the end of the jump, the bishops all stand in different squares of the same row.Find the total number of permutations
of the numbers 
with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order 
, from left to right.Israel
This post has been edited 1 time. Last edited by Assassino9931, Mar 4, 2024, 10:59 AM
through 
.
to 
and move 
rows, so 
for all odd 
by parity, giving a total of 
possible 
from 
,
.
.
is applied one time or isn't applied at all, except for 
of permutations of 
as follows: let 
and 
,
denotes the permutation 
,
consists of 
and 
)
in both of the halves (modulo 
of the permutation we want to achieve is on the left, we just apply one split of size 
that do not change the places of the two halves. Thus, the number of achievable permutations is at least 
for some 
.
.
such that 
for every 
.
here means exists and equals, if it wasn't clear. Note that, because we must have 
,
,
,
,
.
must have preimages, we must have 
,
,
.
must be mapped to itself by 
(and this must exist otherwise by the above logic 
cannot be a bijection), and etc., which implies that 
can be split into intervals of length 
,
,
,
.
,
,
gets mapped to 
,
for all 
(if 
then the result is trivially true). We have 
cases.
and 
,
be the remainder when 
is divided by 
if 
,
,
and 
by the above. Note that 
as 
,
.
,
,
and so 
and so the statement is true in this case.
and 
and 
.
as 
.
,
.
,
so 
and so the statement is true in this case.
,
and 
.
as 
.
,
.
,
so 
and so the statement is true in this case.
and 
as 
.
,
.
,
and so 
and so the statement is true in this case.
can either appear an even number of times and odd number of times in the sequence, corresponding to appearing 
,
(i.e. a jump of length 
,
above; then this will correspond to no moves because we apply 
twice.
horizontally, and the next 
.
bishops: an example with 
is shown below.
Repeating this on the remaining bishops, we find that 
,
.
t
of 
such that there exists a linear combination of the elements of 
,
for 
.
,
,
.
. You will notice that you really can't do this from a forcing algorithm to perform such move (i.e. looking where bishop 
and so on) it must satisfy to cover the entire row so in fact the distance tells the movement of the bishops in "chunks", and since it has an odd prime factor it can't divide 
(you can't achieve 
directly because of the bishops in the middle).
.![\[f:\{1,2,\dots,2^n\}\to\{1,2,\dots,2^n\}\]](http://latex.artofproblemsolving.com/a/5/e/a5e25c2b6d40f6c5c62739e6182850e335ca0774.png)
such that the bishop that is on the 
t
is the same as number of rows that the bishop traveled, so there exists a positive integer 
such that 
.
exists if and only if 
,
means that 
.
to be in the range we must have 
.
,
or 
.
.
.![\[f(\{1,2,\dots,2c\})=\{1,2,\dots,2c\}\]](http://latex.artofproblemsolving.com/b/7/8/b7847ed5bcb25f07e420586107a3d021550a1219.png)
Therefore, when we remove the values from 
,
is a power of two. In each of these sets, the behavior of 
for 
,
,
.
with 
,
.
.
be mapped to each other. Then 
so the behavior of 
and 
,
and ![\[f_i(f_j(x+2^i))=f_j(x)=f_j(f_i(x+2^i))\]](http://latex.artofproblemsolving.com/3/5/f/35fdb6eaf201c7b027240d64fd3e9e97ebf960c4.png)
and less obviously,![\[f_i(f_j(x))=f_j(x)+2^i=f_j(x+2^i)=f_j(f_i(x))\]](http://latex.artofproblemsolving.com/5/a/0/5a03100cbb1c3f023610324bc45ff9c162df6246.png)
where the second equality of the second line is from the fact that 
is used an odd number of times. Since 
for all 
for each used 
