such that ![$1+[\sqrt{2n}]~$](http://latex.artofproblemsolving.com/e/6/a/e6aa1bf5d761758196027add39f2e3cd77db1c1c.png)
divides 
.
, ![$[x]$](http://latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
denotes the largest integer not exceeding 
x
be a scalene triangle with circumcenter 
,
,
.
,
,
at points 
,
,
respectively. Let 
be the projection of 
.
intersects the circumcircle of 
.
and 
intersect 
and 
respectively. Prove that the lines 
,
,
are concurrent.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples 
are integers.
.
Let 
.
be the number of positive integers 
and 
divides
where 
denotes the number of positive integers that divide 
and 
.
and 
then find the value of 
and 
Show that
Let 
Show that
and 
are in the coordinate plane such that 
.
denote the locus of all points 
in the coordinate plane satisfying 
,
be the midpoint of 
.
and 
are on 
and 
.
can be expressed in the form 
,
and 
.
.
Prove that
Where 
are positive integers and 
is a non-square positive integer. If 
denotes the n-th set of solution to the equation with 
Then, prove that 
Y by biomathematics, rkm0959, Kayak, CZRorz, anantmudgal09, integrated_JRC, ThE-dArK-lOrD, Gems98, 62861, tarzanjunior, enhanced, Field_Theory, Math-Ninja, WizardMath, Supercali, AnArtist, mickeydomath, Vrangr, Adventure10, Mango247
For which pairs 
does there exist an injective function 
under which the image of every regular 
gon is a regular 
gon? (Note that 
, and that by a regular 
gon we mean the union of the boundary segments, not the closed polygonal region.)
Sutanay Bhattacharya
does there exist an injective function 
under which the image of every regular 
gon is a regular 
gon? (Note that 
, and that by a regular 
gon we mean the union of the boundary segments, not the closed polygonal region.)Sutanay Bhattacharya
such as each 
is mapped on the right angles of a square. These points determine the backbone of two unique squares.
,
are collinear.
are not collinear. Let 
denote the line pass through 
that the image of them are the same regular 
and 
is not empty set, i.e. WLOG there's a point 
.
must equal to 
for some 
,
.
and (consecutive) sides 
.
and (consecutive) sides 
.
,
that 
.
be the (unique) integer 
,
are two consecutive sides or 
.
.
implies the existence of 
which must contain in both 
and 
,
,
that 
and 
.
.
that 
,
that 
.
must intersect, namely at 
.
.
,
.
that 
,
,
.
with not satisfy any of two properties and also with minimal value of 
.
.
or 
,
and 
are not equal to 
and also 
where 
are two neighbor sides of 
.
and 
.
and 
.
is a cyclic permutation of 
.
,
.
and 
.
,
that 
that there exists 
,
,
.
and 
as its sides, called it 
.
are three consecutive sides and the image of side 
.
sides of 
.
.
