AoPS Academy
that has 8097 distinct points with area of a triangle that has 3 points belong to all 
. Prove that there exists a triangle 
that has the area 
contains at least 2025 points that belong to ( each of that 2025 points can be inside the triangle or lie on the edge of triangle )X
is even it's contradiction. I mean 
.
and 
,we can see the only solution is
and it is contradiction.
and 
. So 
.İt means 
.
but 
.İt is contradiction.
we have 2 option. Firstly 
but 
is contradiction. Secondly 
but is contradiction. I mean 
. So 
. If 
but is contradiction. I mean 
. The only solution is 
.
of positive integers, a partition of into two disjoint nonempty subsets 
and 
is 
if the least common multiple of the elements in is equal to the greatest common divisor of the elements in . Determine the minimum value of 
such that there exists a set of positive integers with exactly 
good partitions.
are lengths of sides of some triangle. Prove the inequality
, where the common ratio 
. Given that
.
. First recall the following fact.
and ratio 
then the sum is given by 
Thus we have that 
As 
thus 
and the finish is easy.
as 
)
Now we just have to find the range of 
as ranges over 
. Over 
the expression takes values in the interval 
, and at 
it takes the value 
, and nowhere else does it take this value. So the answer is 
. 
