[/quote]
,
be positive real numbers such that 
.
,
the number of positive divisors of 
is Burapha integer if it satisfy the following condition
is an odd integer.
holds for every positive divisor 
of 
be real numbers such that 
for all 
.
.
be a quadrilateral with 
.
and 
meet at 
.
and 
denote the circumcircle and the circumcenter of triangle 
.
and 
denote the circumcircle and circumcenter of triangle 
.
meets 
and 
(other than 
and 
)
and 
be the midpoints of minor arcs 
(not including 
(not including 
.
students and a teacher with 
marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of 
.
other students. The same goes on until the teacher is not able to choose anymore student.
and 
be touch points of the incenter of 
at 
and 
,
and 
be the circumcenter of triangles 
and 
,
and 
concurrent.
of one variable that fullfill the following for all real numbers 
and 
:
.
if 
then 
.
,
such that 
.
be the unit circle in the complex plane. Let 
be the map given by 
.
and 
for 
.
is called the period of 
.
of period 
.
)
If the ratio of coffee and cocoa in the mixture 
is 
then the ratio of cocoa in mixture 
which satisfy the following equality for all 
![\[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\]](http://latex.artofproblemsolving.com/2/0/a/20ae7491750bac4c94deaed32d5b9aa66b26491f.png)
Proposed by Dániel Dobák, Budapest
Y by PikaPika999
9Poll:
How was the difficulty of the problems compared to last year?
171 Votes
58%
(100)
25%
(42)
5%
(9)
5%
(9)
2%
(4)
4%
(7)
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I personally thought the problems were much easier than last year, but I didn't really improve as much as I would of liked to improve.
