ortho conf DEF, radius MD, intersect ME,MF, collinear H,K,L
by star-1ord, Mar 23, 2025, 6:05 PM
Let 
be an acute-angled triangle with 
. The altitudes 
and 
intersect at 
. Let 
be the midpoint of 
. Point 
is chosen on the extension of 
beyond and point 
is chosen on the segment 
such that 
. Prove that points 
and are collinear.
be an acute-angled triangle with 
. The altitudes 
and 
intersect at 
. Let 
be the midpoint of 
. Point 
is chosen on the extension of 
beyond and point 
is chosen on the segment 
such that 
. Prove that points 
and are collinear.
be the projection of 
onto 
.
bisects 
.
L
such that 
be an arbitrary point on 
again at 
and 
cuts 
again at 
and 
is 
and 
be the intersection of 
with 
and 
respectively such that 
,
,
are coaxial circles
,
,
,
be 
integers. If 
is an integer, prove that the only solution to
is is 
.
and ![$P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/c/6/4/c64176e52d94032e571b4579e35f9e6adfde675a.png)
such that 
.
such that every equation: 
has no real roots
such that 
,
such that 
such that 
and 
is a perfect square.
so that
such that 
is a perfect square but none of the integers 
are perfect squares.
be a cyclic quadrilateral an let 
The diagonals 
at 
The line through 
mmets the extension of the side 
beyond 
at 
The line through 
meets the extension of the side 
Prove that the circumcircles of the triangles 
and 
are tangent .
May 2018
April 2018