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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1974 Czech and Slovak Olympiad III A
1974 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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Locus problem with bounded areas
Let a unit square
D
\mathcal D
D
be given in the plane. For any point
X
X
X
in the plane denote
D
X
\mathcal D_X
D
X
the image of
D
\mathcal D
D
in rotation with respect to origin
X
X
X
by
+
9
0
∘
.
+90^\circ.
+
9
0
∘
.
Find the locus of all
X
X
X
such that the area of union
D
∪
D
X
\mathcal D\cup\mathcal D_X
D
∪
D
X
is at most 1.5.
5
1
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Areas in cyclic hexagon
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a cyclic hexagon such that AB=BC, CD=DE, EF=FA. Show that
[
A
C
E
]
≤
[
B
D
F
]
[ACE]\le[BDF]
[
A
CE
]
≤
[
B
D
F
]
and determine when the equality holds. (
[
X
Y
Z
]
[XYZ]
[
X
Y
Z
]
denotes the area of the triangle
X
Y
Z
.
XYZ.
X
Y
Z
.
)
4
1
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Uniformly bounded set of polynomials
Let
M
\mathcal M
M
be the set of all polynomial functions
f
f
f
of degree at most 3 such that
∀
x
∈
[
−
1
,
1
]
:
∣
f
(
x
)
∣
≤
1.
\forall x\in[-1,1]:\ |f(x)|\le 1.
∀
x
∈
[
−
1
,
1
]
:
∣
f
(
x
)
∣
≤
1.
Denote
a
a
a
the (possibly zero) coefficient of
f
f
f
at
x
3
.
x^3.
x
3
.
Show that there is a positive number
k
k
k
such that
∀
f
∈
M
:
∣
a
∣
≤
k
\forall f\in\mathcal M:\ |a|\le k
∀
f
∈
M
:
∣
a
∣
≤
k
and find the least
k
k
k
with this property.
3
1
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Sum of permutated digits
Let
m
≥
10
m\ge10
m
≥
10
be any positive integer such that all its decimal digits are distinct. Denote
f
(
m
)
f(m)
f
(
m
)
sum of positive integers created by all non-identical permutations of digits of
m
,
m,
m
,
e.g.
f
(
302
)
=
320
+
023
+
032
+
230
+
203
=
808.
f(302)=320+023+032+230+203=808.
f
(
302
)
=
320
+
023
+
032
+
230
+
203
=
808.
Determine all positive integers
x
x
x
such that
f
(
x
)
=
138
012.
f(x)=138\,012.
f
(
x
)
=
138
012.
2
1
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Max-min of lengths in triangle
Let a triangle
A
B
C
ABC
A
BC
be given. For any point
X
X
X
of the triangle denote
m
(
X
)
=
min
{
X
A
,
X
B
,
X
C
}
.
m(X)=\min\{XA,XB,XC\}.
m
(
X
)
=
min
{
X
A
,
XB
,
XC
}
.
Find all points
X
X
X
(of triangle
A
B
C
ABC
A
BC
) such that
m
(
X
)
m(X)
m
(
X
)
is maximal.
1
1
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Inequality with geometric mean
Let
(
a
k
)
k
=
1
∞
\left(a_k\right)_{k=1}^\infty
(
a
k
)
k
=
1
∞
be a sequence of positive numbers such that
a
k
−
1
a
k
+
1
≥
a
k
2
a_{k-1}a_{k+1}\ge a_k^2
a
k
−
1
a
k
+
1
≥
a
k
2
for all
k
>
1.
k>1.
k
>
1.
For
n
≥
1
n\ge1
n
≥
1
denote
b
n
=
(
a
1
a
2
⋯
a
n
)
1
/
n
.
b_n=\left(a_1a_2\cdots a_n\right)^{1/n}.
b
n
=
(
a
1
a
2
⋯
a
n
)
1/
n
.
Show that also the inequality
b
n
−
1
b
n
+
1
≥
b
n
2
b_{n-1}b_{n+1}\ge b_n^2
b
n
−
1
b
n
+
1
≥
b
n
2
holds for every
n
>
1.
n>1.
n
>
1.