be a function such that for every regular 
-
we have 
.
for all reals 
.
such that 
be positive integers, where 
.
such that
.
.
is the sum of digits of 
taken in base 
)
where 
and 
Prove that
![\[
C(p-1, k-1) \equiv (-1)^{k-1} \pmod{p}
\]](http://latex.artofproblemsolving.com/4/0/7/4072235afbaa1f2d8d4c9ca044615a113a4306ce.png)
for 
,
is the binomial coefficient 
choose 
.
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
,
satisfying
denote all positive divisors of 
Y by Adventure10
Dear Mathlinkers,
1. ABC an acute triangle so that AC < BC < AB
2. B' the midpoint of the segment AC
3. H the orthocenter of ABC
4. D the foot of the A-altitude of ABC
5. O the center of the circumcircle of ABC
6. K the point of intersection of the perpendicular to OD at D with AC.
Prove : HK is perpendicular to B'D.
Sincerely
Jean-Louis
1. ABC an acute triangle so that AC < BC < AB
2. B' the midpoint of the segment AC
3. H the orthocenter of ABC
4. D the foot of the A-altitude of ABC
5. O the center of the circumcircle of ABC
6. K the point of intersection of the perpendicular to OD at D with AC.
Prove : HK is perpendicular to B'D.
Sincerely
Jean-Louis
This post has been edited 1 time. Last edited by jayme, Aug 3, 2016, 10:28 AM
Reason: typo
Reason: typo
be the second intersection of 
with 
.
and 
.
since 
(butterfly), and 
.
.
,
