, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
,
,
terms is divisible by 
be an acute scalene triangle with circumcenter 
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
.
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be two 
complex matrices of the same rank, and let 
be a positive integer. Prove that 
if and only if 
.
to find the third side 
He uses logarithms as follows. He finds 
and doubles it; adds to that the double of 
subtracts the sum of the logarithms of 
and 
divides the result by 
and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?
be a sequence of positive numbers and suppose that 
is divergent. Let further 
.
of positive numbers such that 
is convergent and
.
be given real numbers and let 
for all 
.
converges and find its limit.
.
.
is zero.
of positive integers fast-growing if 
for all 
.
of real numbers and a sequence 
of positive integers, let us denote![\[
S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|.
\]](http://latex.artofproblemsolving.com/2/f/c/2fc080e56fa9d8d6b1cf8a0bb2cb404a65d9ec73.png)
such that for every sequence ![\[
\sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c)
\]](http://latex.artofproblemsolving.com/0/8/a/08aa112d16a9c772145ff03a74fb512b4d02340d.png)
are convergent, then the series 
is also convergent?
such that for every sequence ![\[
S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d)
\]](http://latex.artofproblemsolving.com/5/8/9/589be33c766fde9e227f56d280d02322585e2ba3.png)
are convergent, then the series 
be a ring with 
its centre 
)
the set of invertible elements and 
.
has the property 
if there exists a subgroup 
of group 
such that 
,
and 
for any 
.
has the property 
,
.
and 
are the identity elements for 
and 
does the series 
converge (and absolutely converge)?
greater than 
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.
.
.
pairs of turns where Shayan plays and then Ali plays, Ali can guarantee that 
during their pair of turns. Each move will increase 
.
.
.
.
