[/quote]
,
satisfies the conditions![\[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\]](http://latex.artofproblemsolving.com/5/5/6/556f545540d4941d058a1c4099b4b86eb35caf0d.png)
for all real 
.
such that
for all 
.
,
of rational numbers, where 
are relatively prime positive integers for each 
such that the positive integers 
are all distinct.
![\[A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.\]](http://latex.artofproblemsolving.com/3/d/f/3dfb1d52835627a2977f0074f8df7e7ed677f2a5.png)
can be written as the product of one or more elements of 
,
denote the minimum integer such that 
of integers with 
,
,![\[f(xy)<f(x)+f(y).\]](http://latex.artofproblemsolving.com/c/f/f/cff5554b8e72d99f8989448c2621794242d784d8.png)
(Pairs 
and 
are different if 
or 
)
denote the set of natural numbers, and let 
,
,
.
whose product is a perfect cube.
satisfy the equality, 
prove that the triangle must have a right angle.
,
and 
intersect at 
.
and 
intersect again at 
. If 
, show that 
.
be a positive integer. A graph on 
vertices is given such that the size of the largest clique in the graph is 
,
.
be the midpoint of 
,
be a point inside the triangle such that 
and 
.
,
.
point 
)
.
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
be the unit circle in the complex plane. Let 
be the map given by 
.
and 
for 
.
is called the period of 
.
of period 
.
)
Y by Adventure10, Mango247, Supercali
The Y2K Game is played on a 
grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.
grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.
tiles away from the O and from each end of the grid (the exact number is not important, only that it is far enough from the O that no winning configurations can be achieved next). Then, on B's second turn, he will either win immediately if it is possible, or he can place a second S three boxes away, opposite the direction of A's second move.
These moves will not produce any winning configurations. (Specifically, in the third case, notice that writing an S cannot produce a (S, empty, S) configuration opposite the neighboring S, since that would mean that box 
)
![\[X_1(-)^{a_1}X_2(-)^{a_2}\cdots (-)^{a_n}X_{n+1}\]](http://latex.artofproblemsolving.com/1/c/9/1c91fc301309e35f9cb54f782289ec427d8afc3d.png)
where 
is a string of S and Os that starts and ends with S (except maybe 
and 
that could start and end with O respectively), and has no substring of the form SOS, and 
for all 
.
,
.
.
,
to 
and the second player wins for all even 
structure either by playing two of S or by playing an S in conjunction to Alice's S. Once we have forced such a `trap', noting that Alice cannot play anywhere inside it, we continue to play in safe positions (that is, not `traps') like such
In the end, all that is left is the traps, at which point it must be Alice's move to play (since there are an even number of squares left, 2 per trap) and we are done.
that have exactly two squares between them. We call a square exposed if there exists a neighbor of that square which is not yet filled.
from that square are both unoccupied. On the first player's next turn, he can only fill one of those squares, so the second player can fill the other square, hence building a nuke. 
square, he fills a square adjacent to it with 
or 
.
the second player wrote on:
(or 
but we assume 
,
,
:
(where 
the first 
