[/quote]
,
,
.
points and some chords between them is given.
In every step we can take two chords 
with a common point other than 
and erase exactly one of 
let 
be the minimum of chords after some steps. Find the maximum of 
In every step we can take two chords 
be a positive integer and let 
and 
be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer 
such that at least one of the numbers 
and 
is divisible by 
be the lengths of the sides of a triangle. Prove that![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](http://latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
and 
be arbitrary points on the sides 
and 
of triangle 
,
meets for the second time the circumcircle of 
at point 
.
meets line 
at point 
,
meets line 
at point 
.
and 
pass through the same point.
.
and 
intersect the altitude from 
to 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of 
is the second intersection of circumcircles of triangles 
and 
.
are collinear.
and 
Find the value of 
![$f\in C^1[a,b]$](http://latex.artofproblemsolving.com/8/4/8/848da12534d564560fe1cb016142159234be98b4.png)
,
and suppose that 
,
,
for all ![$x\in [a,b]$](http://latex.artofproblemsolving.com/7/8/d/78d9dcd969a2c45f7cc9234aa1377b6ec1c2d5d3.png)
.
for all 
![$f \in C([0,1];[0,1])$](http://latex.artofproblemsolving.com/4/7/5/475518b7db0b787dd90b93f602c7b912c663fba3.png)
bijective, 
and ![$(a_k) \in [0,1]^\mathbb N$](http://latex.artofproblemsolving.com/a/d/8/ad8a80d847c60ee8e916a1f9b08b7b03ec17c814.png)
with 
converge.
converge?
is called perfect if the sum of its positive divisors 
is twice 
.
is a perfect number since the sum of its positive divisors is 
,![\[
\sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1}
\]](http://latex.artofproblemsolving.com/0/c/0/0c08f527e87f7aae4ca5226f19bd084249551fe3.png)
where the sums are taken over all prime divisors 
of 
Y by
Aman and Berdi, two biologists, they invented a new type of bacteria such that they can control the division of bacteria into several parts. They are also participants of 
with the aim to train for the first problem of . They play the following game.
Initially, they take
bacteria and choose a natural number 
. On each move, the player chooses any 
number from to . Then the player divides each bacterium into pants. Once chosen, the number cannot be chosen twice. If after any player's move the number of bacteria population is divisible by then that player loses. Determine who has the winning strategy depending on the given number if it's known that Amman starts first.
with the aim to train for the first problem of . They play the following game.Initially, they take
bacteria and choose a natural number 
. On each move, the player chooses any 
number from to . Then the player divides each bacterium into pants. Once chosen, the number cannot be chosen twice. If after any player's move the number of bacteria population is divisible by then that player loses. Determine who has the winning strategy depending on the given number if it's known that Amman starts first.
.
,
.
,
.
,
is a prime. If someone choose 
and lose. If B doesn't choose 
numbers less than 
,
,
.
(odd number), so B necessarily has to choose a multiple of 
,
and 
