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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1964 Czech and Slovak Olympiad III A
1964 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
4
1
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Equilateral triangle
Points
A
,
S
A, S
A
,
S
are given in plane such that
A
S
=
a
>
0
AS = a > 0
A
S
=
a
>
0
as well as positive numbers
b
,
c
b, c
b
,
c
satisfying
b
<
a
<
c
b < a < c
b
<
a
<
c
. Construct an equilateral triangle
A
B
C
ABC
A
BC
with the property
B
S
=
b
BS = b
BS
=
b
,
C
S
=
c
CS = c
CS
=
c
. Discuss conditions of solvability.
3
1
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Parametrized quadratic equation
Determine all values of parameter
α
∈
[
0
,
2
π
]
\alpha\in [0,2\pi]
α
∈
[
0
,
2
π
]
such that the equation
(
2
cos
α
−
1
)
x
2
+
4
x
+
4
cos
α
+
2
=
0
(2\cos\alpha-1)x^2+4x+4\cos\alpha+2=0
(
2
cos
α
−
1
)
x
2
+
4
x
+
4
cos
α
+
2
=
0
has 1) a positive root
x
1
x_1
x
1
, 2) if a second root
x
2
x_2
x
2
exists and if
x
2
≠
x
1
x_2\neq x_1
x
2
=
x
1
, the
x
2
≤
0
x_2\leq 0
x
2
≤
0
.
2
1
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Movements on skew lines
Consider skew lines
P
P
′
PP'
P
P
′
,
Q
Q
′
QQ'
Q
Q
′
and points
X
X
X
,
Y
Y
Y
lying on them, respectively. Initially, we have
X
=
P
X=P
X
=
P
,
Y
=
Q
Y=Q
Y
=
Q
. Both points
X
X
X
,
Y
Y
Y
start moving simultaneously along the rays
P
P
′
PP'
P
P
′
,
Q
Q
′
QQ'
Q
Q
′
with the speeds
c
1
c_1
c
1
,
c
2
c_2
c
2
, respectively. Show that midpoint
Z
Z
Z
of segment
X
Y
XY
X
Y
always lies on a fixed ray
R
R
′
RR'
R
R
′
, where
R
R
R
is midpoint of
P
Q
PQ
PQ
.
1
1
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Divisibility by 6000
Show that the number
1
1
100
−
1
11^{100}-1
1
1
100
−
1
is both divisible by
6000
6000
6000
and its last four decimal digits are
6000
6000
6000
.