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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1971 Czech and Slovak Olympiad III A
1971 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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Sum of ratios of lengths in tetrahedron
Let a tetrahedron
A
B
C
D
ABCD
A
BC
D
and its inner point
O
O
O
be given. For any edge
e
e
e
of
A
B
C
D
ABCD
A
BC
D
consider the segment
f
(
e
)
f(e)
f
(
e
)
containing
O
O
O
such that
f
(
e
)
∥
e
f(e)\parallel e
f
(
e
)
∥
e
and the endpoints of
f
(
e
)
f(e)
f
(
e
)
lie on the faces of the tetrahedron. Show that
∑
e
edge
f
(
e
)
e
=
3.
\sum_{e\text{ edge}}\,\frac{\,f(e)\,}{e}=3.
e
edge
∑
e
f
(
e
)
=
3.
5
1
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Locus of vertices of equilateral triangles
Let
A
B
C
ABC
A
BC
be a given triangle. Find the locus
M
\mathbf M
M
of all vertices
Z
Z
Z
such that triangle
X
Y
Z
XYZ
X
Y
Z
is equilateral where
X
X
X
is any point of segment
A
B
AB
A
B
and
Y
≠
X
Y\neq X
Y
=
X
lies on ray
A
C
.
AC.
A
C
.
4
1
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Sum of tangent products
Show that there are real numbers
A
,
B
A,B
A
,
B
such that the identity
∑
k
=
1
n
tan
(
k
)
tan
(
k
−
1
)
=
A
tan
(
n
)
+
B
n
\sum_{k=1}^n\tan(k)\tan(k-1)=A\tan(n)+Bn
k
=
1
∑
n
tan
(
k
)
tan
(
k
−
1
)
=
A
tan
(
n
)
+
B
n
holds for every positive integer
n
.
n.
n
.
3
1
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Partition of positive integers
Consider positive integers
2
,
3
,
…
,
n
−
1
,
n
2,3,\ldots,n-1,n
2
,
3
,
…
,
n
−
1
,
n
where
n
≥
96.
n\ge96.
n
≥
96.
Consider any partition in two (sub)sets. Show that at least one of these two sets always contains two numbers and their product. Show that the statement does not hold for
n
=
95
,
n=95,
n
=
95
,
e.g. there is a partition without the mentioned property.
2
1
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Necessary and sufficient condition for an angle to be right
Let
A
B
C
ABC
A
BC
be a triangle. Four distinct points
D
,
A
,
B
,
E
D,A,B,E
D
,
A
,
B
,
E
lie on the line
A
B
AB
A
B
in this order such that
D
A
=
A
B
=
B
E
.
DA=AB=BE.
D
A
=
A
B
=
BE
.
Find necessary and sufficient condition for lengths
a
=
B
C
,
b
=
A
C
a=BC,b=AC
a
=
BC
,
b
=
A
C
such that the angle
∠
D
C
E
\angle DCE
∠
D
CE
is right.
1
1
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System of inequalities
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
real numbers. Show that there are non-negative
x
,
y
,
z
,
x
y
z
≠
0
x,y,z,xyz\neq0
x
,
y
,
z
,
x
yz
=
0
such that \begin{align*} cy-bz &\ge 0, \\ az-cx &\ge 0, \\ bx-ay &\ge 0. \end{align*}