be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through 
to 
such that 
.
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of 
,
it's circumcenter. Let 
and 
the intersection of the circumcircle of 
,
is the reflection of 
the midpoint of 
,
and the parallel through 
,
and the perpendicular through 
and 
a point on 
,
is the midpoint of 
.
lie on a line.
is inscribed in a circle 
.
— the circle inscribed in a segment 
circle of 
— the centers of circles 
and 
are parallel.
be cyclic quadrilateral. Let 
intersect at 
intersect at 
.
.
with the diagonals of 
and 
are tangent at a fixed point.
,
.
and 
,
is acute, and let 
and 
,
and 
and 
and 
with diameter 
is given. The point 
intersects 
lies on 
is tangent to 
.
,
colours (so that adjacent vertices have different colours).
edges can be removed from the graph so that it remains connected.
be the incenter and the circumcenter of triangle 
be the circumcenters of triangle 
.
be the midpoints of segments 
.
,
.
Prove that
E
matrix equals either 
or 
Let 
denote the sum of eight products of entries in each row. Also, let 
denote the sum of eight products of entries in each column. Find the maximum possible value of 
In other words, find![$$ {\rm max}\ \left[ \sum_{i=1}^8\ \prod_{j=1}^8\ a_{ij} -
\sum_{j=1}^8\ \prod_{i=1}^8\ a_{ij} \right]
$$](http://latex.artofproblemsolving.com/2/d/e/2de7fd40366073445d289356d0f4e66b542cb60c.png)
matrix.
Show that there exists a constant 
such that for all positive integer 
![\[\sum_{k\le n^{\varepsilon}}(n\text{ mod } k)>cn^{2\varepsilon}.\]](http://latex.artofproblemsolving.com/f/e/9/fe933e886e28c2ecf1fa880d0e650ac7a2998aaa.png)
Proposed by Cheng Jiang
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
with 
and 
.
since 
(butterfly), and 
.
.
,
