,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
,
,
lie on a semicircle 
in this order with diameter 
,
.
and 
lie on segments 
and 
respectively such that 
and 
.
passes through 
.
,
are concurrent.
,
of 
points in a plane 
-
points.
and 
are complex numbers such that
complex numbers with 
such that
Find the value of 
.
denote the set of natural numbers and let 
denote 
ordered pairs in 
.
distinct elements in the set 
for 
whose product is a perfect cube.
Where 
are positive integers and 
denotes the n-th set of solution to the equation with 
Then, prove that 
and 
Show that
Let 
Show that
have real roots and real coefficients with 
.
?
,
.
have roots 
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
.
.
.
such that:
Y by
Ali and Shayan are playing a turn-based game on an infinite grid. Initially, all cells are white. Ali starts the game, and in the first turn, he colors one unit square black. In the following turns, each player must color a white square that shares at least one side with a black square. The game continues for exactly 2808 turns, after which each player has made 1404 moves. Let 
be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape by playing optimally. (The perimeter of shape is defined as the total length of the boundary segments between a black and a white cell.)
What are the possible values of the perimeter of, assuming both players play optimally?
be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape by playing optimally. (The perimeter of shape is defined as the total length of the boundary segments between a black and a white cell.)What are the possible values of the perimeter of
, assuming both players play optimally?
.
,
rectangle. Then if we consider the last time occurs then there's at most 
.
to the perimeter and Shayan 
each turn giving the result.
the result follows.
be an index starting at 
.
.
pairs of turns where Shayan plays and then Ali plays, Ali can guarantee that 
.
.
.
.
