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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1973 Czech and Slovak Olympiad III A
1973 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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Length of polygonal chain in square
Consider a square of side of length 50. A polygonal chain
L
L
L
is given in the square such that for every point
P
P
P
of the square there is a point
Q
Q
Q
of the chain with the property
P
Q
≤
1.
PQ\le 1.
PQ
≤
1.
Show that the length of
L
L
L
is greater than 1248.
5
1
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Unknown geometric transformation
Given two points
P
,
Q
P,Q
P
,
Q
of the plane, denote
P
+
Q
P+Q
P
+
Q
the midpoint of (possibly degenerate) segment
P
Q
PQ
PQ
and
P
⋅
Q
P\cdot Q
P
⋅
Q
the image of
P
P
P
in rotation around the origin
Q
Q
Q
under
+
9
0
∘
.
+90^\circ.
+
9
0
∘
.
a) Are these operations commutative? b) Given two distinct points
A
,
B
A,B
A
,
B
the equation
Y
⋅
X
=
(
A
⋅
X
)
+
B
Y\cdot X=(A\cdot X)+B
Y
⋅
X
=
(
A
⋅
X
)
+
B
defines a map
X
↦
Y
.
X\mapsto Y.
X
↦
Y
.
Determine what the mapping is. c) Construct all fixed points of the map from b).
4
1
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Sum with floor function and square roots
For any integer
n
≥
2
n\ge2
n
≥
2
evaluate the sum
∑
k
=
1
n
2
−
1
⌊
k
⌋
.
\sum_{k=1}^{n^2-1}\bigl\lfloor\sqrt k\bigr\rfloor.
k
=
1
∑
n
2
−
1
⌊
k
⌋
.
3
1
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Inequality with arithmetic mean
Let
(
a
k
)
k
=
1
∞
\left(a_k\right)_{k=1}^\infty
(
a
k
)
k
=
1
∞
be a sequence of real numbers such that
a
k
−
1
+
a
k
+
1
≥
2
a
k
a_{k-1}+a_{k+1}\ge2a_k
a
k
−
1
+
a
k
+
1
≥
2
a
k
for all
k
>
1.
k>1.
k
>
1.
For
n
≥
1
n\ge1
n
≥
1
denote
A
n
=
1
n
(
a
1
+
⋯
+
a
n
)
.
A_n=\frac1n\left(a_1+\cdots+a_n\right).
A
n
=
n
1
(
a
1
+
⋯
+
a
n
)
.
Show that also the inequality
A
n
−
1
+
A
n
+
1
≥
2
A
n
A_{n-1}+A_{n+1}\ge2A_n
A
n
−
1
+
A
n
+
1
≥
2
A
n
holds for every
n
>
1.
n>1.
n
>
1.
2
1
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Ex-spheres of tetrahedron
Given a tetrahedron
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
, define an
A
1
A_1
A
1
-exsphere such a sphere that is tangent to all planes given by faces of the tetrahedron and the vertex
A
1
A_1
A
1
and the sphere are separated by the plane
A
2
A
3
A
4
.
A_2A_3A_4.
A
2
A
3
A
4
.
Denote
ϱ
1
,
…
,
ϱ
4
\varrho_1,\ldots,\varrho_4
ϱ
1
,
…
,
ϱ
4
of all four exspheres. Furthermore, denote
v
i
,
i
=
1
,
…
,
4
v_i, i=1,\ldots,4
v
i
,
i
=
1
,
…
,
4
the distance of the vertex
A
i
A_i
A
i
from the opposite face. Show that
2
(
1
v
1
+
1
v
2
+
1
v
3
+
1
v
4
)
=
1
ϱ
1
+
1
ϱ
2
+
1
ϱ
3
+
1
ϱ
4
.
2\left(\frac{1}{v_1}+\frac{1}{v_2}+\frac{1}{v_3}+\frac{1}{v_4}\right)=\frac{1}{\varrho_1}+\frac{1}{\varrho_2}+\frac{1}{\varrho_3}+\frac{1}{\varrho_4}.
2
(
v
1
1
+
v
2
1
+
v
3
1
+
v
4
1
)
=
ϱ
1
1
+
ϱ
2
1
+
ϱ
3
1
+
ϱ
4
1
.
1
1
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Trigonometry of a right triangle
Consider a triangle such that
sin
2
α
+
sin
2
β
+
sin
2
γ
=
2.
\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2.
sin
2
α
+
sin
2
β
+
sin
2
γ
=
2.
Show that the triangle is right.