of the triangle 
is extended beyond 
to 
so that 
.
is extended beyond 
to 
so that 
.
,
segments 
and 
are parallel. Angle bisectors of 
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 point 
at 
.
is tangent to 
,
.
and let 
be reals such that 
Y by
Let 
be a triangle with altitudes 
intersect at orthocenter 
. Let 
be an arbitrary point lies on 
. 
be the reflection point of 
on 
. 
intersects 
at 
. Prove that 
passes through the circumcenter of 
.
be a triangle with altitudes 
intersect at orthocenter 
. Let 
be an arbitrary point lies on 
. 
be the reflection point of 
on 
. 
intersects 
at 
. Prove that 
passes through the circumcenter of 
.
This post has been edited 1 time. Last edited by ken3k06, May 25, 2022, 1:38 AM
,
are symmetric wrt 
,
is tangent to 
.
and 
.
to 
,
and 
lie on 
and 
respectively. Let 
be the reflection of 
on 
,
are concyclic. From here, proving 
is easy.
