AoPS Academy
, 
lies on 
When 
, determine 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)
, and Alice tells Bob whether there exists an ![$i \in [n]$](http://latex.artofproblemsolving.com/e/6/7/e67206e47d5d5875105b339f177a3ab8f5151a5e.png)
such that 
. Bob wins if he ever tells Alice the permutation . Prove that Bob can win the game in at most 
moves.
, let 
be a point on the segment 
and let 
be the intersection point of 
with the diagonal 
. The line perpendicular to the segment through intersects the side 
at point 
. Let 
be a point on the segment 
such that 
. Let 
be the intersection point of the diagonal 
and the segment 
.
and 
are concyclic.
bigger than 
water if the last four digit is 
. Does there exists a water perfect square?
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 
, the equation![\[
x^2+2x-3\underbrace{99\dots 99}_{2m}.
\]](http://latex.artofproblemsolving.com/5/4/d/54d311ebb1f6d92ffc1d2b8f4d6d5de225b349e9.png)
be the set of positive integers less than relatively prime to it. Is there a nice pattern for 
? I know from a Wilson's theorem-style argument that it's either 
or 
, but when is it which?
where 
is a monic polynomial of degree 
satisfying 
where is a polynomial of smallest degree possible satisfying 
and 
, if polynomial satisfies 
where is a polynomial of smallest degree satisfying 
and 
if polynomial satisfies 
is a polynomial of degree 
such that 
Evaluate 
and 
where 
denotes greatest integer function.
,
. Let 
.![\[
\sqrt{x-3}+\sqrt{6-x} \leq p
\]](http://latex.artofproblemsolving.com/3/2/b/32b8bf5c8aa4d7561933e38ee0e2a0480e1a2f07.png)
In which 
, hence find the largest value of 
.
, then jumps in some order on each of the numbers 
exactly once and finally returns with his last jump to . Can the total distance he travelled with these 
jumps be a) 
, b) 
?
be the order in which the frog visited the nine points (from to 
). Then the total distance traveled is:
Since each 
appears twice in the expression for , we deduce that is even. Therefore, it is not possible for to be equal to , solving part 
. to be equal to , it is enough to present the following jump sequence:
with 
and 
and 
.
solutions to 
and 
solutions to 
,
and 
,
and 
.
and 
solutions, respectively.
and four solutions to 
.
.
is this frog's first step, 
is its second step etc. Total distance is 
,
