is extended beyond 
to 
so that 
.
is extended beyond 
to 
so that 
.
,
and 
meet at 
.
and 
meet at 
.
is cyclic
= 2∠AMP.
| 
with 
are positive integers and 
is prime 
)
,![\[
1^2 + 2^2 + 3^2 + \ldots + k^2 = \frac{k(k + 1)(2k + 1)}{6}
\]](http://latex.artofproblemsolving.com/b/5/d/b5d49e25bd8d8a211b72372ff967558e1a8365ce.png)
Find the smallest positive integer 
is divisible by 200.
Let ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
denote the set of natural numbers less than or equal to 
![$f(m,n) = \sum_{(x_1,x_2,x_3, \cdots, x_m) \in [n]^{m}} \frac{x_1}{x_1 + x_2 + x_3 + \cdots + x_m} \binom{n}{x_1} \binom{n}{x_2} \binom{n}{x_3} \cdots \binom{n}{x_m} 2^{\left(\sum_{i=1}^{m} x_i\right)}$](http://latex.artofproblemsolving.com/f/d/f/fdf7b2fcfb960e93d59f3f84cef4372262c69499.png)
be the midpoint of side 
at 
.
is tangent to 
,
.
and let 
be reals such that 
Y by
Given a cyclic quadrilateral 
with 
and 
. Lines 
and 
intersect at 
, and lines 
and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let 
and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to 
if and only if 
divides into two triangles with equal area.
with 
and 
. Lines 
and 
intersect at 
, and lines 
and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let 
and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to 
if and only if 
divides into two triangles with equal area.
direction but not pretty sure with the 
direction, can anyone help me?
is a parallelogram and 
.
iff 
.
and 
.
and 
.![\[ \frac{EX}{GX} = \frac{ES}{SG} = \frac{HT}{TF} = \frac{HY}{FY}. \]](http://latex.artofproblemsolving.com/e/0/c/e0c170e9bf04bfbd87a4e72b3b7f2f1c93858b3b.png)
Again, by the similarity this means ![\[ \frac{AB}{CD} = \frac{DA}{BC} \quad \Rightarrow \quad AB\cdot BC = CD\cdot DA, \]](http://latex.artofproblemsolving.com/c/9/0/c90aaa3c7f32f7fdd231d270e38b43fc934e2ce2.png)
which essentially means the areas of 
are equal.
direction.
