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for which each cell of 
table can be filled with one of the letters 
and 
in such a way that:
, one third are 
and one third are ; and [/*], one third are and one third are .[/*] table are each labelled 
to in a natural order. Thus each cell corresponds to a pair of positive integer 
with 
. For 
, the table has 
diagonals of two types. A diagonal of first type consists all cells for which 
is a constant, and the diagonal of this second type consists all cells for which 
is constant.
be a triangle with incenter , incircle 
and circumcircle 
. Let 
be the midpoints of sides 
, 
, 
and let 
be the tangency points of with and , respectively. Let 
be the intersections of line 
with line 
and line 
, respectively, and let 
be the midpoint of arc 
of . lies on ray 
.
bisects 
.
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective. is bijective if for every integer 
, there exists exactly one integer 
such that 
.
is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer on the board with 
, and on Bob's turn he must replace some even integer on the board with 
. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends. and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
and 
is closest to which integer?
be the real polonomial such that all roots are real and distinct. Prove that there is a rational number 
that all roots of 
are real numbers
![\[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\]](http://latex.artofproblemsolving.com/2/d/2/2d248ce9756f0f09265dd538de3b16b22a944c46.png)
