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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1977 Czech and Slovak Olympiad III A
1977 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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Tetrahedrons in cube
A cube
A
B
C
D
A
′
B
′
C
′
D
′
,
A
A
′
∥
B
B
′
∥
C
C
′
∥
D
D
′
ABCDA'B'C'D',AA'\parallel BB'\parallel CC'\parallel DD'
A
BC
D
A
′
B
′
C
′
D
′
,
A
A
′
∥
B
B
′
∥
C
C
′
∥
D
D
′
is given. Denote
S
S
S
the center of square
A
B
C
D
.
ABCD.
A
BC
D
.
Determine all points
X
X
X
lying on some edge such that the volumes of tetrahedrons
A
B
D
X
ABDX
A
B
D
X
and
C
B
′
S
X
CB'SX
C
B
′
SX
are the same.
5
1
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Colored collinear points
Let
A
1
,
…
,
A
n
A_1,\ldots,A_n
A
1
,
…
,
A
n
be different collinear points. Every point is dyed by one of four colors and every of these colors is used at least once. Show that there is a line segment where two colors are used exactly once and the other two are used at least once.
4
1
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System of symmetric equations
Determine all real solutions of the system \begin{align*} x+y+z &=3, \\ \frac1x+\frac1y+\frac1z &= \frac{5}{12}, \\ x^3+y^3+z^3 &=45. \end{align*}
3
1
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Locus in complex plane
Consider any complex units
Z
,
W
Z,W
Z
,
W
with
Im
Z
≥
0
,
Re
W
≥
0.
\text{Im}\ Z\ge0,\text{Re}\,W\ge 0.
Im
Z
≥
0
,
Re
W
≥
0.
Determine and draw the locus of all possible sums
S
=
Z
+
W
S=Z+W
S
=
Z
+
W
in the complex plane.
2
1
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Construction of a rectangle
The numbers
p
,
q
>
0
p,q>0
p
,
q
>
0
are given. Construct a rectangle
A
B
C
D
ABCD
A
BC
D
with
A
E
=
p
,
A
F
=
q
AE=p,AF=q
A
E
=
p
,
A
F
=
q
where
E
,
F
E,F
E
,
F
are midpoints of
B
C
,
C
D
,
BC,CD,
BC
,
C
D
,
respectively. Discuss conditions of solvability.
1
1
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Points in an open ball
There are given 2050 points in a unit cube. Show that there are 5 points lying in an (open) ball with the radius 1/9.