, 
lies on 
When 
,
is increasing or decreasing
Prove that for any positive number 
,
be a positive integer. Alice and Bob play the following game. Alice considers a permutation 
of the set ![$[n]=\{1,2, \dots, n\}$](http://latex.artofproblemsolving.com/8/f/e/8fea25e7cf72aabc60f48c3b30f4d4818a084ae3.png)
and keeps it hidden from Bob. In a move, Bob tells Alice a permutation 
of ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
,![$i \in [n]$](http://latex.artofproblemsolving.com/e/6/7/e67206e47d5d5875105b339f177a3ab8f5151a5e.png)
such that 
.
moves.
,
be a point on the segment 
and let 
be the intersection point of 
with the diagonal 
.
at point 
.
be a point on the segment 
such that 
.
be the intersection point of the diagonal 
and the segment 
.
and 
are concyclic.
bigger than 
water if the last four digit is 
.
be a positive integer. Prove that![\[
\underbrace{99\dots 99}_{k-1}4\underbrace{00\dots 00}_{k-1}9
\]](http://latex.artofproblemsolving.com/2/f/c/2fc13c37764c96401d76f6288476d7372c17435b.png)
can't be a prime number.
be a positive integer. Solve over 
,![\[
x^2+2x-3\underbrace{99\dots 99}_{2m}.
\]](http://latex.artofproblemsolving.com/5/4/d/54d311ebb1f6d92ffc1d2b8f4d6d5de225b349e9.png)
be the set of positive integers less than 
?
or 
,
with 
and 
and 
.
solutions to 
and 
solutions to 
,
and 
,
and 
.
and 
solutions, respectively.
and four solutions to 
.
where 
is a monic polynomial of degree 
satisfying 
where 
and 
,
and 
if polynomial 
such that 
Evaluate 
and 
where 
denotes greatest integer function.
,
.
.
![\[
\sqrt{x-3}+\sqrt{6-x} \leq p
\]](http://latex.artofproblemsolving.com/3/2/b/32b8bf5c8aa4d7561933e38ee0e2a0480e1a2f07.png)
In which 
,
.
Y by Adventure10, Mango247
in the acute triangle 
.
M is a point in the interior of the segment AC and N is a point on the extension of segment AC such that MN=AC.
let D,E be the feet of perpendiculars from M,N onto lines BC,AB respectively
prove that the orthocentre of lies on circumcircle of 
.M is a point in the interior of the segment AC and N is a point on the extension of segment AC such that MN=AC.
let D,E be the feet of perpendiculars from M,N onto lines BC,AB respectively
prove that the orthocentre of
lies on circumcircle of 
meet at 
.
is 
.
and 
are congruent, so 
,
,
