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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1981 Czech and Slovak Olympiad III A
1981 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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Combinatorial 3D geometry
There are given 11 distinct points inside a ball with volume
V
.
V.
V
.
Show that there are two planes
ϱ
,
σ
,
\varrho,\sigma,
ϱ
,
σ
,
both containing the center of the ball, such that the resulting spherical wedge has volume
V
/
8
V/8
V
/8
and its interior contains none of the given points.
5
1
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Discrete optimization
Let
n
n
n
be a positive integer. Determine the maximum of the sum
x
1
+
⋯
+
x
n
x_1+\cdots+x_n
x
1
+
⋯
+
x
n
where
x
1
,
…
,
x
n
x_1,\ldots,x_n
x
1
,
…
,
x
n
are non-negative integers satisfying the condition
x
1
3
+
⋯
+
x
n
3
≤
7
n
.
x_1^3+\cdots+x_n^3\le7n.
x
1
3
+
⋯
+
x
n
3
≤
7
n
.
4
1
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Combinatorial number theory
Let
n
n
n
be a positive integer. Show that there is a prime
p
p
p
and a sequence
(
a
k
)
k
≥
1
\left(a_k\right)_{k\ge1}
(
a
k
)
k
≥
1
of positive integers such that the sequence
(
p
+
n
a
k
)
k
≥
1
\left(p+na_k\right)_{k\ge1}
(
p
+
n
a
k
)
k
≥
1
consists of distinct primes.
3
1
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Locus of vertices of equilateral triangles
Let
A
B
C
D
ABCD
A
BC
D
be a unit square. Consider an equilateral triangle
X
Y
Z
XYZ
X
Y
Z
with
X
,
Y
X,Y
X
,
Y
as (inner or boundary) points of the square. Determine the locus
M
M
M
of vertices
Z
Z
Z
of all these triangles
X
Y
Z
XYZ
X
Y
Z
and compute the area of
M
.
M.
M
.
2
1
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Intersection of collinear segments
Let
n
n
n
be a positive integer. Consider
n
2
+
1
n^2+1
n
2
+
1
(closed, i.e. including endpoints) segments on a single line. Show that at least one of the following statements holds: a) there are
n
+
1
n+1
n
+
1
segments with non-empty intersection, b) there are
n
+
1
n+1
n
+
1
segments among which two of them are disjoint.
1
1
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Parametrized inequality
Determine all
a
∈
R
a\in\mathbb R
a
∈
R
such that the inequality
x
4
+
x
3
−
2
(
a
+
1
)
x
2
−
a
x
+
a
2
<
0
x^4+x^3-2(a+1)x^2-ax+a^2<0
x
4
+
x
3
−
2
(
a
+
1
)
x
2
−
a
x
+
a
2
<
0
has at least one real solution
x
.
x.
x
.