and 
be the incircle and circumcircle of the acute triangle 
so that all of its sides are tangent to 
,
are both on 
and 
,
and 
,
and 
intersects on 
be integers greater than 
,
be positive integers not greater than 
.
not greater than 
,![\[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \]](http://latex.artofproblemsolving.com/7/4/1/7414caaacc9fd761961279f8a37ab6cd6af6b49a.png)
where 
denotes the greatest common divisor of 
.
be a fixed integer. Let 
and for all 
let 
.
for which sequence 
will be bounded?
denotes Euler’s Phi Function, meaning that 
gives the number of integers within the set 
that are relative primes to 
bounded if there exist a constant 
for all positive integers 
be a non-constant polynomial with integer coefficients such that 
.
to be the set of positive divisors of 
is 
-![$$f[\text{divs}(m)]=\text{divs}(n).$$](http://latex.artofproblemsolving.com/1/2/3/12341f07a9f4396dbced8a380259613f2399b112.png)
Prove that for any such ![$f[S]$](http://latex.artofproblemsolving.com/f/0/1/f01f35bea4da88837806e6056a6e416673778c48.png)
for some set 
denotes the set 
.
Y by son7
A point 
is marked on altitude 
of triangle 
. Line 
intersects side 
at point 
, line 
intersects side 
at point 
. Points 
and 
are projections of points and on side 
, respectively. The area of the triangle 
is twice the area of the triangle 
.In what ratio does the altitude divide 
?
is marked on altitude 
of triangle 
. Line 
intersects side 
at point 
, line 
intersects side 
at point 
. Points 
and 
are projections of points and on side 
, respectively. The area of the triangle 
is twice the area of the triangle 
.In what ratio does the altitude divide 
?
bisects the angle 
.
,
and 
