They fill in real numbers to empty spaces (the coefficients of the terms other than 
and the constant term) in turn. If the resulting polynomial has no real root, Mr. Fat wins; otherwise, Mr. Taf wins. If Mr. Fat goes first, who has a winning strategy?
.
array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A move consists of choosing a row or a column and changing the color of every cell in the chosen row or column.
different positive integers , less than 
,
Y by
On the sides 
and 
of the triangle 
we chose the points 
and 
, respectively, so that 
. Let 
be the point of intersection of the lines 
and 
the foot of the perpendicular dropped from the point on the line 
, and 
is a point of the segment such that 
. Prove that the inscribed circle of triangle 
touches the side at the point .
(Nagel Igor)
and 
of the triangle 
we chose the points 
and 
, respectively, so that 
. Let 
be the point of intersection of the lines 
and 
the foot of the perpendicular dropped from the point on the line 
, and 
is a point of the segment such that 
. Prove that the inscribed circle of triangle 
touches the side at the point .(Nagel Igor)
This post has been edited 1 time. Last edited by parmenides51, Jul 14, 2020, 9:58 PM
it suffice to show that the 
-
touches the side 
at the point 
.
be a point such that 
.
are isosceles the 
are extrernal bisectors of 
P
are collinear.This follows from the fact that 
is the orthocenter of 
.
