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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1969 Czech and Slovak Olympiad III A
1969 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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Tangent circles on a sphere
A sphere with unit radius is given. Furthermore, circles
k
0
,
k
1
,
…
,
k
n
(
n
≥
3
)
k_0,k_1,\ldots,k_n\ (n\ge3)
k
0
,
k
1
,
…
,
k
n
(
n
≥
3
)
of the same radius
r
r
r
are given on the sphere. The circle
k
0
k_0
k
0
is tangent to all other circles
k
i
k_i
k
i
and every two circles
k
i
,
k
i
+
1
k_i,k_{i+1}
k
i
,
k
i
+
1
are tangent for
i
=
1
,
…
,
n
i=1,\ldots,n
i
=
1
,
…
,
n
(assuming
k
n
+
1
=
k
1
k_{n+1}=k_1
k
n
+
1
=
k
1
). a) Find relation between numbers
n
,
r
.
n,r.
n
,
r
.
b) Determine for which
n
n
n
the described situation can occur and compute the corresponding radius
r
.
r.
r
.
(We say non-planar circles are tangent if they have only a single common point and their tangent lines in this point coincide.)
5
1
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Locus in plane
Two perpendicular lines
p
,
q
p,q
p
,
q
and a point
A
∉
p
∪
q
A\notin p\cup q
A
∈
/
p
∪
q
are given in plane. Find locus of all points
X
X
X
such that
X
A
=
∣
X
p
∣
⋅
∣
X
q
∣
,
XA=\sqrt{|Xp|\cdot|Xq|\,},
X
A
=
∣
Xp
∣
⋅
∣
Xq
∣
,
where
∣
X
p
∣
|Xp|
∣
Xp
∣
denotes the distance of
X
X
X
from
p
.
p.
p
.
4
1
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Set of complex numbers given by inequality
Determine all complex numbers
z
z
z
such that
∣
z
−
∣
z
+
∣
z
∣
∣
∣
−
∣
z
∣
3
≥
0
\Bigl|z-\bigl|z+|z|\bigr|\Bigr|-|z|\sqrt3\ge0
z
−
z
+
∣
z
∣
−
∣
z
∣
3
≥
0
and draw the set of all such
z
z
z
in complex plane.
3
1
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Infinite sequences with given property
Let
p
p
p
be a prime. How many different (infinite) sequences
(
a
k
)
k
≥
0
\left(a_k\right)_{k\ge0}
(
a
k
)
k
≥
0
exist such that for every positive integer
n
n
n
a
0
a
1
+
a
0
a
2
+
⋯
+
a
0
a
n
+
p
a
n
+
1
=
1
?
\frac{a_0}{a_1}+\frac{a_0}{a_2}+\cdots+\frac{a_0}{a_n}+\frac{p}{a_{n+1}}=1?
a
1
a
0
+
a
2
a
0
+
⋯
+
a
n
a
0
+
a
n
+
1
p
=
1
?
2
1
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Bounded are of a convex quadrilateral
Five different points
O
,
A
,
B
,
C
,
D
O,A,B,C,D
O
,
A
,
B
,
C
,
D
are given in plane such that
O
A
≤
O
B
≤
O
C
≤
O
D
.
OA\le OB\le OC\le OD.
O
A
≤
OB
≤
OC
≤
O
D
.
Show that for area
P
P
P
of any convex quadrilateral with vertices
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
(not necessarily in this order) the inequality
P
≤
1
2
(
O
A
+
O
D
)
(
O
B
+
O
C
)
P\le \frac12(OA+OD)(OB+OC)
P
≤
2
1
(
O
A
+
O
D
)
(
OB
+
OC
)
holds and determine when equality occurs.
1
1
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Equation in Q[\sqrt5]
Find all rational numbers
x
,
y
x,y
x
,
y
such that
(
x
+
y
5
)
2
=
7
+
3
5
.
\left(x+y\sqrt5\right)^2=7+3\sqrt5.
(
x
+
y
5
)
2
=
7
+
3
5
.