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differents subjects, each consist of 
students. Given that for any 
different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least 
different subjects.
+1 w
be positive real numbers such that 
. Prove that 
, denote by 
the number of positive divisors of . We say that a positive integer 
is Burapha integer if it satisfy the following condition
is an odd integer.
holds for every positive divisor 
of , such that 
be real numbers such that 
for all 
.
.
be a quadrilateral with 
. Diagonals 
and 
meet at 
. Let 
and 
denote the circumcircle and the circumcenter of triangle 
. Let 
and 
denote the circumcircle and circumcenter of triangle 
. Segment 
meets and again at 
and 
(other than 
and 
), respectively. Let 
and 
be the midpoints of minor arcs 
(not including ) and 
(not including ). Prove that 
.
and 
be touch points of the incenter of 
at 
and 
, respectively. Let 
and 
be the circumcenter of triangles 
and 
, respectively. Show that 
and 
concurrent.
of one variable that fullfill the following for all real numbers 
and 
:
.
be a nonempty set of positive integers such that:
if 
then 
. for any prime 
, there exists 
such that 
. is finite.
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer such that 
is called the period of 
. Determine the total number of points in 
of period 
.
)
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
has side lengths 
,
,
.
,
,
and denote them as 
,
,
.
?
