AoPS Academy
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be a set of integers with the following properties:
.
and 
, then 
.
, 
is composite, then all positive divisors of are in . contains all positive integers.
for all 
.
be an odd positive integer satisfying 
with exactly 2 distinct prime factors. Prove that is a prime number or the square of a prime number.
that satisfy the following condition: For every integer k such that 
there always exists a positive integer 
that is divisible by n and 
. 
: sum of elements of
is drawn externally on side 
of 
. Point 
is taken on 
. Find 
if 
.
and 
lie in a straight line in that order. Suppose that 
is a point not on the line and that 
and 
Find the area of 
to 
is 24 from herons on 
Gives 
Then, we know 
so 
therefore 
so 
By Heron's we get the area of 
has the same height as 
so the area of 
to this segment. Several AIME problems have shown me triangles with nice side lengths. For this one, the cute triangle was 
. You can get it from taking the colloquial 
, and then removing a 
.
and find 
, 
, 
, and 
in terms of 
. Find using 
. Then use Heron's on 
and multiply by 
to get ![\[168\cdot \frac{39}{14}=\boxed{468}.\]](http://latex.artofproblemsolving.com/e/2/b/e2b9aba05c08b1bdc4c4069860dedfcaccac222a.png)
has 
,
,
.
be the angle bisector of angle 
such that 
.
and 
.
?
of the numbers 
satisfy 
and 
?
as well using herons formula to find the height.
.
such that the intersection is between C and D (we will call that . So, by herons, we have
.
and 
.
,
.
