MathDB
Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1957 Czech and Slovak Olympiad III A
1957 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
2
1
Hide problems
Construction and calculation in a pyramid
Consider a (right) square pyramid
A
B
C
D
V
ABCDV
A
BC
D
V
with the apex
V
V
V
and the base (square)
A
B
C
D
ABCD
A
BC
D
. Denote
d
=
A
B
/
2
d=AB/2
d
=
A
B
/2
and
φ
\varphi
φ
the dihedral angle between planes
V
A
D
VAD
V
A
D
and
A
B
C
ABC
A
BC
. (1) Consider a line
X
Y
XY
X
Y
connecting the skew lines
V
A
VA
V
A
and
B
C
BC
BC
, where
X
X
X
lies on line
V
A
VA
V
A
and
Y
Y
Y
lies on line
B
C
BC
BC
. Describe a construction of line
X
Y
XY
X
Y
such that the segment
X
Y
XY
X
Y
is of the smallest possible length. Compute the length of segment
X
Y
XY
X
Y
in terms of
d
,
φ
d,\varphi
d
,
φ
. (2) Compute the distance
v
v
v
between points
V
V
V
and
X
X
X
in terms of
d
,
φ
.
d,\varphi.
d
,
φ
.
4
1
Hide problems
Construction of a trapezoid
Consider a non-zero convex angle
∠
P
O
Q
\angle POQ
∠
POQ
and its inner point
M
M
M
. Moreover, let
m
>
0
m>0
m
>
0
be given. Construct a trapezoid
A
B
C
D
ABCD
A
BC
D
satisfying the following conditions: (1) vertices
A
,
D
A, D
A
,
D
lie on ray
O
P
OP
OP
and vertices
B
,
C
B,C
B
,
C
lie on ray
O
Q
OQ
OQ
, (2) diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect in
M
M
M
, (3)
A
B
=
m
AB=m
A
B
=
m
. Prove that your construction is correct and discuss conditions of solvability.
3
1
Hide problems
Integer values of cotangent function
Find all real numbers
α
\alpha
α
such that both values
cot
(
α
)
\cot(\alpha)
cot
(
α
)
and
cot
(
2
α
)
\cot(2\alpha)
cot
(
2
α
)
are integers.
1
1
Hide problems
Easy equation with a parametr
Find all real numbers
p
p
p
such that the equation
x
2
−
5
p
2
=
p
x
−
1
\sqrt{x^2-5p^2}=px-1
x
2
−
5
p
2
=
p
x
−
1
has a root
x
=
3
x=3
x
=
3
. Then, solve the equation for the determined values of
p
p
p
.