[/quote]
,
be an acute triangle with incenter 
,
,
.
be the midpoint of 
.
meets 
at 
.
and 
the circumcircles of 
and 
,
meets 
and 
,
meets 
and 
.
and 
.
and 
,
and 
of 
.
on segment 
is tangent to 
.
meets 
.
be a point on 
be a circle thaat passes through 
,
be a circle that passes through 
and tangents to 
is tangent to circles 
and 
.
,
.
,
lines intersect at point 
,
= 2∠AMP.
I
be positive real numbers, show that:
such that 
and 
are also primes.
be pairwise distinct integers. Show that the equation 
has at most one integral solution.
be a positive integer and let 
be odd primes. Prove that the positive integers 
can be colored in 
colors, such that for any 
of the same color, 
is not divisible by 
.
be the Gergonne point of the triangle 
is parallel with line 
.
be a polynomial with real coefficients such that 
for all 
.
for some real constant 
and 
.
.
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
,
such that 
and 
are both divisible by 
over the positive integers as follows:
where 
such that 
.
such that 
and 
is a perfect square.
or 
and 
and 
operated by two different companies, then there must exist flight route 
operated by the third company different from 
,
and 
denote the number of flight routes operated by companies 
.
Y by Adventure10, Mango247
Alexander has chosen a natural number 
and has written down in a line,and in increasing order,all his positive divisors 
(where 
and 
).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these 
numbers (the greatest common divisors) is equal to 
.Find all possible values of 
.
and has written down in a line,and in increasing order,all his positive divisors 
(where 
and 
).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these 
numbers (the greatest common divisors) is equal to 
.Find all possible values of 
.
my beloved
only which clearly works. We now prove that we must have 
.
for all 
,
Thus for the sum of the written numbers is 
everywhere except for one index, where 
.
,
are odd. Thus there are two "consecutive" factors of 
and 
.
,
and 
are also consecutive and different from our original pair. But we have
which becomes clear if we multiply everything by 
,
so equality does not hold here either.
,
,
and 
is the least prime divisor of 
,
,
as desired. 
only, which works.
,
except in one instance where 
are both odd and 
.
unless 
must be 
;
and 
.
So 
which is an obvious contradiction as 
is odd. Hence 
.
which is 
is bounded by 
The second 
is bounded by 
Continuing in this manner, the sum is bounded by 
Since 
it suffices to check 
Clearly, only 
is a value 
by euclidean algorithm. Then, the sum is at most 
Thus, there exists exactly one value such that 
Hence, 
Thus, 
must also be odd values. Hence, 
However, observe that 
whereas 
where we also lose values. Thus, 
By earlier, we are done.
