[/quote]
,
is given. Let 
and 
be the midpoints of 
and 
,
on segment 
is given. Draw equilateral 
such that 
is on 
is on 
.
and all real numbers 
:
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
so that the following relation holds for all 
.
and 
such that 
.
are positive isn't necessary.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
.
and 
intersect 
and 
respectively.
to 
.
is extended beyond 
to 
.
,
be a cyclic quadrilateral. Let 
and 
and the lines 
and 
meet at 
.
and 
liters of milk. Then, 
liters are removed from the container and replaced with 
Y by
Another window has a design which is shown below. The whole window is a square; the
central section is made of plain glass; the outer section is coloured. The curved lines are
arcs of circles of the same radius as the side length of the square and centred on its
corners. What is the ratio of coloured to plain glass in the whole window? Give your
answer in the form 1 : ?, where ? is a decimal correct to 3 sf.
central section is made of plain glass; the outer section is coloured. The curved lines are
arcs of circles of the same radius as the side length of the square and centred on its
corners. What is the ratio of coloured to plain glass in the whole window? Give your
answer in the form 1 : ?, where ? is a decimal correct to 3 sf.
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