AoPS Academy
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.
such that for all 
we have:
be an acute triangle inscribed in its circumcircle 
, and let 
be its incircle. Let 
be the point where the 
incircle of triangle 
touches 
. Suppose line 
intersects segment 
at 
, and intersects line 
at 
. Let the line through 
perpendicular to intersect line 
at 
. Prove that: ![\[AI \parallel PA'.\]](http://latex.artofproblemsolving.com/0/2/6/0264e64b5edff4aca1e35037c5b0f86763c3b4e7.png)
be a point in the interior of a tetrahedron 
. Show that the sum of the angles 
, 
, 
, 
, 
, 
is greater than 
be a point in the interior of a tetrahedron . Show that the sum of the angles , , , , , is greater than
then prove that 
