of cardboard, with 5 rectangular faces instead of 6 and there is no leftover cardboard. What is the largest volume this box can hold given that 
of cardboard is needed to cover the box?
Subject to the constraints:
Prove that
and 
Prove that
. Prove that
can be written in the form![\[
\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},
\]](http://latex.artofproblemsolving.com/9/e/9/9e96c07ea3b0750af8bed344df6e8a86fcf0d772.png)
where 
and 
are positive integers. Find the value of 
.
be an acute triangle with 
,
.
be the altitude from vertex 
.
and 
be the feet of the perpendiculars from 
to the lines 
and 
,
intersect the circle 
and 
such that 
.
be the midpoint of segment 
intersect line 
at point 
.
.
is parallel to line 
.
be the orthocenter of triangle 
,
be the midpoint of segment 
is the center (HBC). Let point 
lie on line 
such that ray 
is the angle bisector of 
.
intersect at a point lying on circle 
(i.e., the square on 
has 
,
,
to the centers of the opposite squares are concurrent.
be positive integers. Maximize 
if 
.
,
,
.
Y by
The midpoint of each side of the base of the quadrangular pyramid is connected with a segment to the intersection point of the medians of the opposite side face. Prove that:
a) these segments intersect and the intersection point are divided in a ratio of
, counting from the side of the foundation;
b) the midpoints of these segments are vertices of parallelogram.
Find the ratio of the area of this parallelogram to the area the base of the pyramid.
a) these segments intersect and the intersection point are divided in a ratio of
, counting from the side of the foundation;b) the midpoints of these segments are vertices of parallelogram.
Find the ratio of the area of this parallelogram to the area the base of the pyramid.
This post has been edited 1 time. Last edited by parmenides51, Jul 11, 2020, 6:15 AM
