`1
Find the volume of this prism
`2
Find the ratio of the lateral area of
this prism to the area of its base
`3
Find the total area of a cube whose
volume is equal to the volume of this
prism
`4
This is a regular prism. Find the dis-
tance from vertex F to line AE
`5
This is a regular prism. Find the dis-
tance from vertex D to line CE
`6
Cut this prism by a plane which passes
through C, divides edge DE in half
and edge FE in the ratio 3:1 and
find the area of section
`7
Cut this regular prism by a plane pa-
rallel to its bases, so that the to-
tal area of the prism is divided in
the ratio 2:1 when considering from the
lower base
`8
Cut this regular prism by a plane which
contains edge AC and makes an angle
of 70 degrees with the base
`9
Cut this regular prism with the bisec-
tor plane of the dihedral angle between
base DEF and face BCFE
`10
Construct on edge FC of this prism a
point from which edge AB is seen
at an angle of 50 degrees
`11
Cut this cube by a plane which passes
through edge EF and divides the volume
of the cube in ratio 7:3 when conside-
ring from the lower base
`12
Cut this cube through the midpoints of
edges HE, EF and HD. Find the perimeter
of the section
`13
Cut this cube along edge AE and perpen-
dicularly to plane CFH and find the
area of the section
`14
Cut this cube by a plane which passes
through vertices B and H perpendicular-
ly to face ABFE
`15
Cut this cube by plane HFA and find the
distance from point E to this
plane
`16
Cut this cube by plane AFC and find the
distance from point H to this
plane
`17
BC is a base of an isosceles trapezoid
which is equivalent to any face of this
cube. The shortest base of the trapezo-
id lies on edge EH. Construct it
`18
Cut this cube by a plane which passes
through its center, divides edge DH in
the ratio 3:1 and edge AB in the ratio
5:3
`19
Construct on edge AE of this cube a
point from which edge HG is seen at
an angle of 40 degrees
`20
Cut this cube by a plane which passes at
right angles to EB, below the center of
the cube and cut off a prism whose vo-
lume makes up 3/4 from the volume of
the cube
`21
Find the volume of this pyramid
`22
Find the volume of this pyramid
`23
This regular pyramid is sectioned by
a plane containing a side of the base.
Calculate the minimal area of the
section
`24
Find out the shortest path from A to C
via edge DB
`25
Find the aria of the section of this
pyramid by a plane which divides
edges DC and AB in half and edge CA
in ratio 1:2
`26
Cut this regular pyramid through verti-
ces B and C, so that the ratio of the
area of the section to the area of the
base is equal to 7:6
`27
Construct the center of the sphere cir-
cumscribed about this regular pyramid
`28
Construct the center of the sphere in-
scribed in this regular pyramid
`29
Cut this pyramid by a plane which is
parallel to the base and divides its
volume in the ratio 2:3 when conside-
ring from the vertex
`30
Construct on edge DB of this regular
pyramid a point from which edge AC
is seen at an angle of
55 degrees
`31
Find the volume of this parallelipiped
`32
The top and the bottom are removed from
this box, so that it becomes flexible.
Then its walls are positioned at right
angles to each other, and a new, rec-
tangular, bottom is glued to. How many
times will thus the volume of the box
enlarge?
`33
Calculate the edge of a cube whose to-
tal area is equal to the total area of
this parallelipiped
`34
Cut this parallelipiped by a plane
which contains edge CD and makes an
angle of 20 degrees with the base
`35
Find the angle between edge CG and
plane BEH
`36
Cut this parallelipiped by a plane
which divides edge DH in half and
is parallel to plane EBC
`37
This right parallelipiped is sectioned
by a plane which contains edge BF and
is perpendicular to plane EGC. Find
the area of the section
`38
Construct on edge HD of this paralleli-
piped a point which is equidistant from
vertices B and E
`39
Build the shortest way from vertex F of
this right parallelipiped to vertex D
via edge EH
`40
Find the distance between straight
lines AF and HC in this parallelipiped
`41
Find the total area of this paralleli-
piped
`42
Find the angle between straight lines
HC and BG in this parallelipiped
`43
Cut off from this rectangular paralle-
lipiped a triangular pyramid whose 
three vertices are A, F and E, and the
volume is 10 times smaller as the pa-
rallelipiped's
`44
Construct on segment FH a point whose
distance from vertex A is equal to the
longest edge of this parallelipi-
ped
`45
Find the projection of vertex H of this
parallelipiped on segment GA
`46
This rectangular parallelipiped is cut
through the midpoints of edges HG, HE
and AB. Find the area of the section
`47
Section this parallelipiped by a plane
that bisects edges CD and BF and is
parallel to line BD
`48
Section this rectangular parallelipiped
by a plane which is perpendicular to
diagonal BH and divides the paralleli-
piped into two congruent bodies
`49
This rectangular parallelipiped is sec-
tioned by a plane that contains one of
diagonals of the lower base and is pa-
rallel to one of diagonals of the pa-
rallelipiped. Calculate the volume of
the pyramid obtained
`50
This rectangular parallelipiped is sec-
tioned by a plane that bisects edge GF
and is perpendicular to straight line
BD. Find the volume of the lesser body
obtained
`51
Find the volume of this regular pyra-
mid
`52
Find the perimeter of the section of
this regular pyramid by a plane that
passes through a side of its base per-
pendicularly to the opposite face
`53
Find the angle between the opposite la-
teral faces of this regular pyramid
`54
Through two opposite vertices of this
regular pyramid a plane is drawn, such
that the area of the section is mini-
mal. Find the distance from the vertex
of the pyramid to this plane
`55
Cut this regular pyramid by a plane
which edge EC is perpendicular to and
vertices B, A and D are equidistant
from
`56
Construct on edge ED of this regular
pyramid a point, different from D,
from which segment AC is seen at the
same angle as from point D
`57
Through the midpoint of edge EB of this
regular pyramid a perpendicular to edge
EA plane is drawn. Find the area of the
section obtained
`58
Through the ortocenter of face AED of
this regular pyramid a straight line
perpendicular to this face is drawn.
Construct the point of intersection of
this line with the base of the pyramid
`59
Cut this regular pyramid by a plane
parallel to the base and tangent to the
sphere inscribed in the pyramid
`60
Cut this regular pyramid by a plane
which is parallel to face ABE and
passes through the center of the sphere
circumscribed about the pyramid
`61
Find the volume of this truncated pyra-
mid
`62
Cut this regular truncated pyramid
along edge AC, so that the area of sec-
tion is equal to the area of one of
its bases
`63
On face ABED a point is taken, such
that the sum of its distances from the
vertices of this face is minimal.
Through this point a perpendicular line
to the same face is drawn.Construct the
second point of intersection of this
line with the surface of the pyramid
`64
What is the angle at which edge DE is
seen from vertex C?
`65
We draw a plane parallel to the alti-
tude of this regular truncated pyramid
and equidistant from the vertices of
the upper base of the pyramid. Find the
area of the section
`66
Cut this regular truncated pyramid by a
plane that does not contain any of ver-
tices and divides the volume of the py-
ramid in half
`67
Cut this truncated pyramid parallel to
edge AD and so that the section is
an isosceles triangle
`68
Cut this regular truncated pyramid
along one of its edges, so that
the section is an equilateral
triangle
`69
Find the radius of the sphere circum-
scribed about this truncated pyramid
`70
Find the maximal area of a sections of
this regular truncated pyramid by a
plane parallel to one of its lateral
faces
`71
Find the volume of this prism
`72
Find the total area of this prism
`73
Find the distance from vertex F of this
prism to the plane of the opposite
lateral face
`74
Find in face ABED a point which is
projected into the center of face ACFD
`75
Section this prism in such a way
that the section is an isosceles
triangle
`76
Cut this prism by a plane equidistant
from straight lines ED and CF
`77
Point out the vertex of the greatest
angle of section EDC of this prism
`78
Construct on segment AE a point from
which segment AC is seen at an right
angle
`79
Find the minimal area of the section of
this prism by a plane which intersects
the three lateral edges
`80
On edge ED an equidistant from A and
C point lies. What is the angle at
which BC is seen from that point?
`81
Find the maximal area of triangular
sections of this cube
`82
Find the angle between plane ABF and
the plane drawn through F, A and the
midpoint of edge  HE
`83
The surface of a sphere is equivalent
to the surface of this cube. How many
times the volume of the sphere is lar-
ger as the cube's?
`84
This cube is sectioned through verti-
ces F,B and the midpoint of edge HE.
From the center of the section a per-
pendicular line to its plane is erec-
ted. How long the part of the line
surrounded by the cube is?
`85
Cut this cube by a plane which is pa-
rallel to plane ACF but is three times
nearer to vertex H as ACF is, and find
the area of the section
`86
Vertex F and the midpoint of edge AB
are symmetric with respect to a plane.
Cut the cube by this plane
`87
Construct the center of a sphere which
contains the vertices of the lower base
and is tangent to the upper one
`88
Cut this cube through the midpoint of
edge AD and parallel to HC and BF
`89
Construct on diagonal HF a point from
which diagonal AH is seen at an angle
of 100 degrees
`90
Find the radius of a sphere tangent to
all edges of this cube
`91
Draw in the plane of face DCGH a
straight line parallel to AF and as far
from E as the length of AF is
`92
From the midpoint of edge DC a perpen-
dicular line to plane HFA is dropped.
Construct the second point of intersec-
tion of this line with the surface of
the cube
`93
We inscribe a sphere in this cube then,
in the space bounded by the sphere's
and the cube's surfaces, a new sphere
of maximal radius. Find this radius
`94
This cube is sectioned through D,C and
the midpoint of edge AE. A cylinder is
inscribed in the triangular prism ob-
tained. Find its volume
`95
A plane is drawn which passes through
H,A, intersects edge EF and makes an
angle of 70 degrees with it. How far is
vertex C from this plane?
`96
Find the maximal length of segments
surrounded by the surface of this cube
and perpendicular to one of its diago-
nals
`97
This cube is sectioned parallel to
plane GBE, so that edge AD is bisected.
Find the area of the section
`98
Section this cube perpendicularly to
segment AF, so that the section is the
same area as any of cube's
faces is
`99
Section this cube by a plane that pas-
ses through vertices D and A, between
vertices F and B and as far from edge
HE as long a half of diagonal of the
cube is
`100
Through C and D a plane which divides
edge BF in the ratio 1:2 is drawn. Find
the angle between it and the diagonals
of the upper base
`101
Find the volume of this regular trun-
cated pyramid
`102
Find the distance from vertex A to
straight line HF
`103
A segment is perpendicular to line CH
and joins vertex A to a point of edge
HD. Find the length of the
segment
`104
This regular truncated pyramid is sec-
tioned through a vertex of the lesser
base parallel to one of opposite late-
ral faces. Find the volume of the les-
ser body resulted
`105
Through vertex H of this regular trun-
cated pyramid a straight line parallel
to plane AEB is drawn, so that its seg-
ment surrounded by the surface of the
pyramid is as long as possible.
Construct this segment
`106
On the base of this regular truncated
pyramid, inside it a sphere of maximal
size is situated. Cut the pyramid
through the points where the sphere
touches its lateral surface
`107
Cut this regular truncated pyramid pa-
rallel to line AE, so that the sec-
tion is a rectangle
`108
Cut this regular truncated pyramid by a
plane which is parallel to face AEHD
and equidistant from edges FG
and BC
`109
This is a regular truncated pyramid.
Find the angle between straight lines
HD and AF
`110
Build the shortest way from
vertex B to vertex H via
edge AE
`111
From vertex E a perpendicular line to
plane HFA is dropped. Find the length
of its segment surrounded by the sur-
face of the parallelipiped
`112
This rectangular parallelipiped is sec-
tioned through two opposite vertices of
the upper base, so that its volume is
divided in the ratio 1:5. Find the
perimeter of the section
`113
This rectangular parallelipiped is sec-
tioned by a plane parallel to plane
HDB and as far from it as long a half
of the shortest edge of the paralleli-
piped is. Find the perimeter of the
section
`114
A sphere is circumscribed about this
rectangular parallelipiped. Then the
parallelipiped is pulled away. What is
the volume of the body remained?
`115
Find the maximal area of the section of
this rectangular parallelipiped by a
plane perpendicular to the bases
`116
Cut this right parallelipiped by a
plane perpendicular to the bases, so
that the section is of maximal peri-
meter
`117
Cut this parallelipiped in such a way
that the section is a square
`118
Cut this parallelipiped by a plane pa-
rallel to edge CD, so that the section
is a rectangle
`119
Cut this right parallelipiped through
one of its vertices, so that the sec-
tion is a rectangular triangle
`120
Construct on the surface of this paral-
lelipiped a point at equal distances
from A,F and at minimal distance from H
`121
Find the volume of this regular
pyramid
`122
Find the angle between lateral adjacent
faces of this regular pyramid
`123
This regular pyramid is sectioned along
a side of the base perpendicularly to
the opposite lateral face. Find the
distance from the center of the base to
the plane of the section
`124
This regular pyramid is sectioned
through the center of the base parallel
to one of lateral faces. Find the area
of the section
`125
In this regular pyramid a cylinder is
inscribed which is twice lower as the
pyramid. Calculate the diameter of the
base of the cylinder
`126
Cut this regular pyramid parallel to
the base, so that the section is equi-
valent to a lateral face
`127
Cut this regular pyramid parallel to
edge GD, so that the section is an
isosceles triangle
`128
A ray is drawn from the vertex B of
this regular pyramid, so that it makes
an angle of 25 degrees with the base
and intersects the surface of the pyra-
mid in a point of the opposite lateral
edge. Construct this point
`129
Cut this regular pyramid into two bo-
dies of the same volume, so that the
section does not contain any of pyra-
mid's vertices
`130
Which of vertices of this regular pyra-
mid is the furthest from A?
`131
Find the maximal area of those secti-
ons of this regular prism which are
perpendicular to its base
`132
Through vertex D of this regular prism
straight lines parallel to plane BCI
are drawn. What is the maximal length
of their segments surrounded by the
prism's surface?
`133
A sphere and a cylinder are circum-
scribed about this regular prism. Cal-
culate the ratio of their volumes
`134
Find the distance from vertex I of this
regular prism to vertex E
`135
Find the angle between straight lines
JE and HC in this regular prism
`136
Draw through vertex B, in the plane of
face ABHG a straight line which is per-
pendicular to line FK
`137
Cut this regular prism along edge FL,
behind the center and so that the vo-
lume of the prism is divided in the
ratio 1:8
`138
Cut this regular prism parallel to its
altitude, so that the section is a
square
`139
In this regular prism a regular trian-
gular prism is inscribed, so that its
vertices bisects some of the edges of
the base of the given prism. Find the
ratio of the total areas of the prisms
`140
Find the area of the section of this
regular prism by a plane that contains
D and is parallel to plane LGB
`141
Find the length of the altitude of this
pyramid
`142
We inscribe a cylinder in this pyramid,
so that one of its base lies on face
ABC, and its altitude is twice short-
er as the pyramid's. Find the total
area of the cylinder
`143
Find the greatest dihedral angle of
this pyramid
`144
This pyramid is sectioned by the plane
of symmetry of vertices A and C.Find
the volume of the cut off triangular
pyramid
`145
Cut this pyramid, so that the section
is a parallelogram
`146
Cut this pyramid parallel to its base,
so that the area of the section is
three times smaller as the area of
the base
`147
Find the perimeter of the section of
this pyramid by a plane which is paral-
lel to plane ADC and passes through the
center of gravity of face ABD
`148
Construct on face BCD of this pyramid
a segment, so that all its points are
equidistant from vertices
A and B
`149
Find the distance between straight
lines AD and BC
`150
Through A, in the plane of face ABD a
line is drawn perpendicularly to the
median of face BDC (drawn from B). Con-
struct the point of intersection of
this line with edge BD
`151
Find the volume of this truncated pyra-
mid
`152
Find the distance from vertex B of this
truncated pyramid to plane ADC
`153
Find the area of the section of this
truncated pyramid by a plane that pas-
ses through C perpendicularly to edge
AB
`154
Find the perimeter of the section of
this truncated pyramid by a plane that
divides edge EB in the ratio 1:2 and is
parallel to plane CDF
`155
This truncated pyramid is laid upon the
table, so that it is held up at its
edge AB. What is the maximal altitude
the pyramid can rise above the
table?
`156
Cut this truncated pyramid by a plane
which is parallel to edge EF and di-
vides in half the volume of the py-
ramid
`157
Construct in the plane of the upper
base of this truncated pyramid a line
parallel to AB and as far from it as
the length of edge AD is
`158
Section this truncated pyramid along
edge FE, so that the section is a
parallelogram
`159
Construct on the lower base of this
truncated pyramid a point from which
edge FD is seen at a maximal angle
`160
Cut this truncated pyramid perpendicu-
larly to edge AB, so that the
section is a triangle of
maximal area
`161
Find the volume of this prism
`162
Through vertex H straight lines perpen-
dicular to FG are drawn. What is the
maximal length of their segments sur-
rounded by the prism's surface?
`163
From this block of wood shaped like
a prism is to be made a cylindrical
piece just as tall as the prism. What
is the maximal volume of this
piece?
`164
Cut this right prism along edge GC, so
that the area of the section is
maximal
`165
Cut this right prism along any of its
edges, so that the volume of the prism
is divided in half
`166
Through edge DH of this right prism a
plane perpendicular to the plane of
face FBCG is drawn. Draw the segment
which is the section of this face
`167
Through C a plane parallel to plane BED
is drawn. Find the point of its inter-
section with edge HG
`168
Join vertices A and E on the surface of
this prism by a broken line whose all
points are equidistant from the plane
of face FBCG
`169
This right prism is held up by a table
at its vertex F, so that another vertex
rises as much as possible above the
table. Point out this vertex
`170
Cut this prism through its vertex G, so
that the section is a rectangular tri-
angle
`171
All edges of this polyhedron are con-
gruent. Calculate its volume
`172
Find the maximal area of the sections
of this polyhedron that pass through FD
`173
This polyhedron is held up by table
on one of its base's edge. How high
can the polyhedron rise above the
table?
`174
Find the greatest of the dihedral
angles of this polyhedron
`175
This polyhedron whose all edges are
congruent is sectioned by a plane that
is perpendicular to one of the edges of
the base and divides this edge in the
ratio 1:3. Find the volume of the les-
ser polyhedron resulted
`176
This polyhedron whose all edges are
congruent is sectioned through ver-
tex F by a plane parallel to plane
DEG. Find the perimeter of the section
`177
All edges of this bottomless box are
congruent. We upset the box then embed
in it a ball of maximal size. What is
the distance from the center of the
ball to the initial top of the box?
`178
Cut this polyhedron by a plane that is
equidistant from its vertex and the
plane of its base
`179
Cut this polyhedron by a plane that
does not contain any of its vertices 
and divides its volume in half
`180
Construct on face ABED a point which
is equidistant from those faces that
join together in vertex F
`181
Find the greatest dihedral angle of
this polyhedron
`182
Cut this polyhedron along edge AB, so
that its total area is divided in the
ratio 1:2 when considering from
vertex E
`183
From which point belonging to this po-
lyhedron whose all edges are congruent
the median of face ACE drawn to side
AC is seen at a minimal angle?
`184
All edges of this polyhedron are con-
gruent. Draw through the center of face
ABE a perpendicular line to DC
`185
Through the midpoint of edge BD a line
parallel to edge EB is drawn. Construct
the second point of intersection of
this line with the surface of the poly-
hedron
`186
A sphere is inscribed in this polyhe-
dron whose all edges are congruent. A
plane touches the sphere, is parallel
to plane ABC and intersects the upper
part of the polyhedron. Cut the poly-
hedron by this plane
`187
A maximal size spherical cavity has
been formed inside this block. Find the
volume of the material remained
`188
A cube is the same total area as this
polyhedron. How many times the volume
of the cube is greater as the polyhe-
dron's?
`189
Cut this polyhedron parallel to edge
AB, so that the section is a
rhombus
`190
Cut this polyhedron whose all edges are
congruent by a plane that does not con-
tain any of vertices, so that the sec-
tion is a parallelogram
`191
Calculate the volume of this regular
octahedron
`192
Find the distance between two opposite
faces of this regular octahedron
`193
Find the area of the section of this
regular octahedron by the plane of sym-
metry of vertices C and F
`194
This regular octahedron is sectioned
parallel to two of its opposite faces
and equidistant from them. Find the
perimeter of the section
`195
Find the area of the sphere circum-
scribed about this regular octahedron
`196
Cut the upper part of this regular oc-
tahedron by a plane parallel to plane
BAD, so that the section is the same
area as any of faces
`197
Draw in this regular octahedron through
F, in plane FAB a straight line that
makes with the bisector of angle DCE
an angle of 70 degrees
`198
Cut this regular octahedron perpendicu-
larly to edge FD, so that edge AE is
divided in the ratio 2:3
`199
Construct the point where face FAB of
this regular octahedron touches the
sphere inscribed
`200
Construct the projection of the mid-
point of edge EB of this regular octa-
hedron on plane AFD
`201
This regular pyramid is situated be-
tween two parallel planes. What is the
minimal distance between planes?
`202
This pyramid stands on one of its
lateral faces. What is the minimal
altitude at which the pyramid could
rise?
`203
This metalic framework has to be drag-
ged through a wide and low tunnel. What
minimal height the tunnel should have?
`204
From this wooden cube a cylindrical 
cork is made whose diameter of base is
a quarter of the cube's edge. What is
the maximal length of the top?
`205
Suppose you join vertices C and D of 
this prism by a three-side broken line
which also passes through other two
distinct vertices of the prism. Find
the maximal length of the broken line
`206
Cut this right parallelipiped by a
plane that contains one of its edges
so that the section is of maximal
area
`207
A cube is cut off from this paralleli-
piped. What is the maximal volume of
the cube?
`208
What is the maximal volume of spheres
contained in this parallelipiped?
`209
A beetle moved from vertex D of this
regular pyramid in a straight line un-
til it got edge AE, then, in the same
manner, to vertex B. So it covered a
distance which is twice the altitude
of the pyramid. Draw the beetle's
path
`210
This aquarium contains a certain
amount of water. Somebody introduces
a metalic ball whose diameter is equal
to the altitude of the container, so
that the water gets its upper edge.
Mark on BH the initial level of the
water
`211
This parallelipiped is situated between
two parallel planes each having at
least 2 common points with it. Find the
maximal distance between the planes
`212
Two planes are symmetric with respect
to the center of this rectangular pa-
rallelipiped, also are they parallel to
the shortest edge of the base, and make
an angle of 25 degrees with the base.
They trisect the volume of the paralle-
lipiped. Find the distance between them
`213
Edge EF of this rectangular paralleli-
piped is the axis of a cylinder and AB
is one of its generatrices. Find the
volume of that part of the parallelipi-
ped which is out of cylinder
`214
7/8 of capacity of this container was
filled by water. Somebody introduced
in it, one by one, several heavy balls
whose diameter is a quarter of the
shortest edge of the container, until
the water overflew the container. How
many balls were introduced?
`215
The interior surface of this rectangu-
lar parallelipiped is made of mirrors.
A ray of light starts from vertex D at
equal angles to the three edges. Const-
ruct the second point of reflection of
the ray from the interior
surface
`216
A straight line passes through vertex C
of this regular pyramid and makes con-
gruent angles with all edges that join
together in the same vertex. Find the
second point of intersection of the
line with the surface of the pyramid
`217
A straight line passes through vertex
B of this regular pyramid and makes
congruent angles with all faces that
join together in the same vertex. Find
the second point of intersection of the
line with the surface of the pyramid
`218
Construct on edge CB of this regular
pyramid a point which is equidistant
from vertices A and E
`219
Cut this pyramid by a plane which is
equidistant from vertices B and D, is
perpendicular to the base of the pyra-
mid, and divides its volume in the 
ratio 1:10 when considering from A
`220
A beetle starts from vertex F of
this regular pyramid, gets as soon as
possible edge AG then, in the same
manner, edge BG etc., until it re-
turns to edge FG. How high is the
beetle above the base of the pyramid
now?
`221
Find the area of the sphere tangent to
the planes of all lateral faces of this
regular pyramid and to all edges of its
base
`222
Find the maximal radius of a sphere
surrounded by this framework
`223
Find the volume of the pyramid cut off
from this body by a plane which passes
through a vertex of the base perpendi-
cularly to the opposite lateral edge
`224
A cone is inscribed in this regular py-
ramid. Its vertex belongs to edge ED
and its base lies on face ABG. Find the
maximal volume of the cone
`225
A cylindrical bar is passed through the
"holes" of this metalic framework. What
is the maximal area of the circular
cross-section of the bar?
`226
Cut this prism parallel to straight
line CD, so that the volume of the
prism is divided in half
`227
Cut this prism, so that the section is
a rectangle
`228
Cut this prism, so that the section is
a rhombus that does not contain any of
vertices of the prism
`229
Cut this prism, so that the section is
a square
`230
Cut this prism, so that the section is
an isosceles triangle that contains
one of the vertices of the
prism
`231
Cut this prism, so that the section is
a square that does not contain any
of the vertices of the
prism
`232
Find the maximal perimeter of triangu-
lar sections of this polyhedron
`233
This block of glass is molten, then
poured into a spherical form. How many
times does the total area of the block
decrease?
`234
This polyhedron is situated between 2
parallel planes. What is the minimal
distance between the planes?
`235
All edges of this polyhedron are con-
gruent. An ant moved, along the
shortest way, from vertex B to edge
EC then, along the shortest way
again, to vertex D via edge AC.
Build the ant's
path 
`236
This building is shaped as a polyhe-
dron whose all edges are congruent. Cut
away its lower part parallel to plane
ACB, so that the building could hold
the ceiling of a hall whose height is
equal to the length of any edge of the
polyhedron
`237
Cut this regular octahedron by a plane
which does not contain any of its
vertices and bisects its
volume
`238
Find the maximal volume of a sphere
which could be surrounded by this
steel framework
`239
From two wires of the same length two
different frameworks have been made:
one shaped as a cube and the other as
a regular octahedron below. How many
times the volume of the cube is greater
than the octahedron's?
`240
This toy is shaped as a regular octahe-
dron.Somebody fixed a thread in a point
that divides one of the edges in the
ratio 1:2, and so hung the toy from the
ceiling. Then he measured both the ma-
ximal and the minimal heights above the
floor. Calculate their difference
`241
Cut this cube by a plane that divides
the volume of the cube in half, is
equidistant from vertices A and F, is
twice nearer to E as to B, and does
not intersect face ABFE
`242
A cylindrical bar is passed through two
opposite "holes" of this framework.
Find the maximal area of the circular
cross-section of the bar
`243
A long bar with maximal square cross-
section is passed through "holes" DEC
and BAE of this framework, so that one
of the bar's faces is close to AB. Find
the point where the bar touches BE
`244
This wooden block is drilled across, so
that both bases are holed, all lateral
faces remain intact, and the cross-
section of the hole is maximal. Find
the volume of the body remained
`245
Four segments of minimal length are
removed from this framework, then re-
placed by four others, so that the
framework becomes a parallelipiped
of maximal altitude. How many times
does the height of the framework
increase?
`246
In this wooden block a cylindrical hole
with axis of revolution EC is drilled,
so that edges HG and GF remain intact.
Calculate the maximal radius of the
hole
`247
Vertices H and E of this regular prism
are centers of bases of a cylinder. The
lateral surface of the cylinder touches
edge CI in some point. Construct this
point
`248
A long cylindrical bar is passed
through this framework, so that the la-
teral surface of the bar touches all
edges of the pyramid, excepting BC.
Construct the point where it touches
edge AD
`249
Three bars of minimal length are re-
moved from this framework, they being
used for prolonging the lateral bars.
Then the framework is reshaped like
a triangular pyramid. Find its
altitude
`250
A sphere is tangent to all edges of the
lower base of this regular truncated
pyramid and to edge AE. Find the radi-
us of the sphere
