AoPS Academy
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.
given that there exist positive integers 
satisfying
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 
.
be an odd prime number. How many -element subsets 
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 
. 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 .
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: 
. 
.
denote the number of positive integers less than that are relatively prime to . Prove that there exists a positive integer for which the equation 
has at least 
solutions in .
that can be written in the form
where 
are pairwise coprime positive integers such that 
, 
, and 
.
, 
, 
, 
are integers such that 
.
and 
of 
such that for each integer 
with 
, we have![\[ n\mid a_i - b_i - c_i
\]](http://latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
be a finite abelian group of order 
.
be arbitrary.
and pairwise distinct 
such that 
for 
.
and 
s.t 
and 
and there are exactly two i;j s.t 
and 
.Then if we know two permutation good for the sequence (a_1...a_n) then we can build two permuttionm good for (b_1...b_n)
Please post it here and now!
that add 
to 
and subtract from 
.
that add to and subtract from .
and 
of ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
for some sequence ![$\{ a_i\}_{i\in [n]}$](http://latex.artofproblemsolving.com/f/6/e/f6e384c04b4aaf07513dacb66f8cadc0a597069e.png)
so that 
for all ![$i\in [n]$](http://latex.artofproblemsolving.com/e/e/5/ee51c5c5eb09111cc774a00e9970074ad9f9058c.png)
.![$\{ b_i\}_{i\in [n]}$](http://latex.artofproblemsolving.com/1/3/e/13e91e327fa3f2bc2d43edbe110540d350184040.png)
with sum divisible by that differ, in modulo , from in only two positions, say 
and 
.
and 
of so that 
for all .
for all that 
.
by, for each integer 
, define ![$i_{k+1}\in [n]$](http://latex.artofproblemsolving.com/c/e/5/ce54cff3726a2dae0e8cc97f7b358dd90622aeee.png)
to be the unique integer satisfy 
.
are the indices that 
with minimal 
, and then minimal 
.
. This means 
for all 
. for 
gives us
Plugging and use 
gives us 
.
, contradiction to the definition of 
.
. Let 
. Define the desired permutations and as follows:
and 
for the rest that 
.
is just to not use 
more than one time.
for all except when 
. and are permutations of , we've ![$\sum_{i\in [n]}{(\sigma' (i)+\tau' (i))} \equiv_n 2\times \frac{n(n-1)}{2}\equiv_n 0\equiv_n \sum_{i\in [n]}{b_i}$](http://latex.artofproblemsolving.com/4/0/c/40ce5067c15701668f9fc9a82369ce8ab1f4384c.png)
.
when too. This prove the validity of permutations we constructed.
.
