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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1955 Czech and Slovak Olympiad III A
1955 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
2
1
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Cuboid between spheres
Let
S
1
,
S
2
\mathsf{S}_1,\mathsf{S}_2
S
1
,
S
2
be concentric spheres with radii
a
,
b
a,b
a
,
b
respectively, where
a
<
b
.
a<b.
a
<
b
.
Denote
A
B
C
D
A
′
B
′
C
′
D
′
ABCDA'B'C'D'
A
BC
D
A
′
B
′
C
′
D
′
a square cuboid (
A
B
C
D
,
A
′
B
′
C
′
D
ABCD,A'B'C'D
A
BC
D
,
A
′
B
′
C
′
D
are the squares and
A
A
′
∥
B
B
′
∥
C
C
′
∥
D
D
′
AA'\parallel BB'\parallel CC'\parallel DD'
A
A
′
∥
B
B
′
∥
C
C
′
∥
D
D
′
) such that
A
,
B
,
C
,
D
∈
S
2
A,B,C,D\in\mathsf{S}_2
A
,
B
,
C
,
D
∈
S
2
and the plane
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
is tangent to
S
1
.
\mathsf{S}_1.
S
1
.
Finally assume that
A
B
A
A
′
=
a
b
.
\frac{AB}{AA'}=\frac ab.
A
A
′
A
B
=
b
a
.
Compute the lengths
A
B
,
A
A
′
.
AB,AA'.
A
B
,
A
A
′
.
How many of such cuboids exist (up to a congruence)?
1
1
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Trapezoid
Consider a trapezoid
A
B
C
D
,
A
B
∥
C
D
,
A
B
>
C
D
.
ABCD,AB\parallel CD,AB>CD.
A
BC
D
,
A
B
∥
C
D
,
A
B
>
C
D
.
Let us denote intersections of lines as follows:
E
=
A
C
∩
B
D
,
F
=
A
D
∩
B
C
.
E=AC\cap BD, F=AD\cap BC.
E
=
A
C
∩
B
D
,
F
=
A
D
∩
BC
.
Let
G
H
GH
G
H
be a line such that
G
∈
A
D
,
H
∈
B
C
,
E
∈
G
H
,
G
H
∥
A
B
.
G\in AD,H\in BC, E\in GH,GH\parallel AB.
G
∈
A
D
,
H
∈
BC
,
E
∈
G
H
,
G
H
∥
A
B
.
Moreover, denote
K
,
L
K,L
K
,
L
midpoints of the bases
A
B
,
C
D
AB,CD
A
B
,
C
D
respectively. Show that (a) the points
K
,
L
K,L
K
,
L
lie on the line
E
F
,
EF,
EF
,
(b) lines
A
C
,
K
H
AC,KH
A
C
,
KH
and
B
D
,
K
G
BD,KG
B
D
,
K
G
are not parallel (denote
M
=
A
C
∩
K
H
,
N
=
B
D
∩
K
G
M=AC\cap KH,N=BD\cap KG
M
=
A
C
∩
KH
,
N
=
B
D
∩
K
G
), (c) the points
F
,
M
,
N
F,M,N
F
,
M
,
N
are collinear.
3
1
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Counting vertices
In the complex plane consider the unit circle with the origin as its center. Furthermore, consider inscribed regular 17-gon with one of its vertices being
1
+
0
i
.
1+0i.
1
+
0
i
.
How many of its vertices lie in the (open) unit disc centered in
3
/
2
(
1
+
i
)
\sqrt{3/2}(1+i)
3/2
(
1
+
i
)
?
4
1
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Real root of equation
Given that
a
,
b
,
c
a,b,c
a
,
b
,
c
are distinct real numbers, show that the equation
1
x
−
a
+
1
x
−
b
+
1
x
−
c
=
0
\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}=0
x
−
a
1
+
x
−
b
1
+
x
−
c
1
=
0
has a real root.