lying outside the circle 
,
and 
.
is chosen on the segment 
.
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
be an acute-angled triangle with 
and let 
be the foot of the
-
.
and 
in line 
and 
respectively. A line through 
and 
respectively such that 
while 
and 
and 
are tangent to each other.
with circumcircle 
.
is the angle bisector of 
.
intersects 
.
angle bisector of 
if and only if 
.
be real numbers satisfying
Y by Adventure10, Mango247
A trapezium is given with parallel bases having lengths 
and 
. Split it into two trapeziums by a cut, parallel to the bases, of length 
. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in 
and 
trapeziums, respectively, so that all the 
trapezoids obtained have the same area. Determine the minimum possible value for and the lengths of the cuts to be made to achieve this minimum value.
and 
. Split it into two trapeziums by a cut, parallel to the bases, of length 
. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in 
and 
trapeziums, respectively, so that all the 
trapezoids obtained have the same area. Determine the minimum possible value for and the lengths of the cuts to be made to achieve this minimum value.
a
