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Problems
Contests
International Contests
Hungary-Israel Binational
1993 Hungary-Israel Binational
1993 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(7)
7
1
Hide problems
on |G'| = 2 (G' the commutator subgroup of G)
In the questions below:
G
G
G
is a finite group;
H
≤
G
H \leq G
H
≤
G
a subgroup of
G
;
∣
G
:
H
∣
G; |G : H |
G
;
∣
G
:
H
∣
the index of
H
H
H
in
G
;
∣
X
∣
G; |X |
G
;
∣
X
∣
the number of elements of
X
⊆
G
;
Z
(
G
)
X \subseteq G; Z (G)
X
⊆
G
;
Z
(
G
)
the center of
G
;
G
′
G; G'
G
;
G
′
the commutator subgroup of
G
;
N
G
(
H
)
G; N_{G}(H )
G
;
N
G
(
H
)
the normalizer of
H
H
H
in
G
;
C
G
(
H
)
G; C_{G}(H )
G
;
C
G
(
H
)
the centralizer of
H
H
H
in
G
G
G
; and
S
n
S_{n}
S
n
the
n
n
n
-th symmetric group. Assume
∣
G
′
∣
=
2
|G'| = 2
∣
G
′
∣
=
2
. Prove that
∣
G
:
G
′
∣
|G : G'|
∣
G
:
G
′
∣
is even.
6
1
Hide problems
ab^2 = b^3a and ba^2 = a^3b => a=b=1
In the questions below:
G
G
G
is a finite group;
H
≤
G
H \leq G
H
≤
G
a subgroup of
G
;
∣
G
:
H
∣
G; |G : H |
G
;
∣
G
:
H
∣
the index of
H
H
H
in
G
;
∣
X
∣
G; |X |
G
;
∣
X
∣
the number of elements of
X
⊆
G
;
Z
(
G
)
X \subseteq G; Z (G)
X
⊆
G
;
Z
(
G
)
the center of
G
;
G
′
G; G'
G
;
G
′
the commutator subgroup of
G
;
N
G
(
H
)
G; N_{G}(H )
G
;
N
G
(
H
)
the normalizer of
H
H
H
in
G
;
C
G
(
H
)
G; C_{G}(H )
G
;
C
G
(
H
)
the centralizer of
H
H
H
in
G
G
G
; and
S
n
S_{n}
S
n
the
n
n
n
-th symmetric group. Let
a
,
b
∈
G
.
a, b \in G.
a
,
b
∈
G
.
Suppose that
a
b
2
=
b
3
a
ab^{2}= b^{3}a
a
b
2
=
b
3
a
and
b
a
2
=
a
3
b
.
ba^{2}= a^{3}b.
b
a
2
=
a
3
b
.
Prove that
a
=
b
=
1.
a = b = 1.
a
=
b
=
1.
5
1
Hide problems
H \leq G, |H | = 3
In the questions below:
G
G
G
is a finite group;
H
≤
G
H \leq G
H
≤
G
a subgroup of
G
;
∣
G
:
H
∣
G; |G : H |
G
;
∣
G
:
H
∣
the index of
H
H
H
in
G
;
∣
X
∣
G; |X |
G
;
∣
X
∣
the number of elements of
X
⊆
G
;
Z
(
G
)
X \subseteq G; Z (G)
X
⊆
G
;
Z
(
G
)
the center of
G
;
G
′
G; G'
G
;
G
′
the commutator subgroup of
G
;
N
G
(
H
)
G; N_{G}(H )
G
;
N
G
(
H
)
the normalizer of
H
H
H
in
G
;
C
G
(
H
)
G; C_{G}(H )
G
;
C
G
(
H
)
the centralizer of
H
H
H
in
G
G
G
; and
S
n
S_{n}
S
n
the
n
n
n
-th symmetric group. Let
H
≤
G
,
∣
H
∣
=
3.
H \leq G, |H | = 3.
H
≤
G
,
∣
H
∣
=
3.
What can be said about
∣
N
G
(
H
)
:
C
G
(
H
)
∣
|N_{G}(H ) : C_{G}(H )|
∣
N
G
(
H
)
:
C
G
(
H
)
∣
?
4
2
Hide problems
rooks on a 3n x 3n chessboard
Find the largest possible number of rooks that can be placed on a
3
n
×
3
n
3n \times 3n
3
n
×
3
n
chessboard so that each rook is attacked by at most one rook.
|aH\cap Hb| and |H|, here H\leq G
In the questions below:
G
G
G
is a finite group;
H
≤
G
H \leq G
H
≤
G
a subgroup of
G
;
∣
G
:
H
∣
G; |G : H |
G
;
∣
G
:
H
∣
the index of
H
H
H
in
G
;
∣
X
∣
G; |X |
G
;
∣
X
∣
the number of elements of
X
⊆
G
;
Z
(
G
)
X \subseteq G; Z (G)
X
⊆
G
;
Z
(
G
)
the center of
G
;
G
′
G; G'
G
;
G
′
the commutator subgroup of
G
;
N
G
(
H
)
G; N_{G}(H )
G
;
N
G
(
H
)
the normalizer of
H
H
H
in
G
;
C
G
(
H
)
G; C_{G}(H )
G
;
C
G
(
H
)
the centralizer of
H
H
H
in
G
G
G
; and
S
n
S_{n}
S
n
the
n
n
n
-th symmetric group. Let
H
≤
G
H \leq G
H
≤
G
and
a
,
b
∈
G
.
a, b \in G.
a
,
b
∈
G
.
Prove that
∣
a
H
∩
H
b
∣
|aH \cap Hb|
∣
a
H
∩
H
b
∣
is either zero or a divisor of
∣
H
∣
.
|H |.
∣
H
∣.
3
2
Hide problems
five points on a semicircle with radius 1
Distinct points
A
,
B
,
C
,
D
,
E
A, B , C, D, E
A
,
B
,
C
,
D
,
E
are given in this order on a semicircle with radius
1
1
1
. Prove that
A
B
2
+
B
C
2
+
C
D
2
+
D
E
2
+
A
B
⋅
B
C
⋅
C
D
+
B
C
⋅
C
D
⋅
D
E
<
4.
AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.
A
B
2
+
B
C
2
+
C
D
2
+
D
E
2
+
A
B
⋅
BC
⋅
C
D
+
BC
⋅
C
D
⋅
D
E
<
4.
elements of n-th symmetric group
In the questions below:
G
G
G
is a finite group;
H
≤
G
H \leq G
H
≤
G
a subgroup of
G
;
∣
G
:
H
∣
G; |G : H |
G
;
∣
G
:
H
∣
the index of
H
H
H
in
G
;
∣
X
∣
G; |X |
G
;
∣
X
∣
the number of elements of
X
⊆
G
;
Z
(
G
)
X \subseteq G; Z (G)
X
⊆
G
;
Z
(
G
)
the center of
G
;
G
′
G; G'
G
;
G
′
the commutator subgroup of
G
;
N
G
(
H
)
G; N_{G}(H )
G
;
N
G
(
H
)
the normalizer of
H
H
H
in
G
;
C
G
(
H
)
G; C_{G}(H )
G
;
C
G
(
H
)
the centralizer of
H
H
H
in
G
G
G
; and
S
n
S_{n}
S
n
the
n
n
n
-th symmetric group. Show that every element of
S
n
S_{n}
S
n
is a product of
2
2
2
-cycles.
1
2
Hide problems
a/b is written in the decimal system as b.a, find a and b
Find all pairs of coprime natural numbers
a
a
a
and
b
b
b
such that the fraction
a
b
\frac{a}{b}
b
a
is written in the decimal system as
b
.
a
.
b.a.
b
.
a
.
from (xy)^i = x^iy^i show that G is Abelian
In the questions below:
G
G
G
is a finite group;
H
≤
G
H \leq G
H
≤
G
a subgroup of
G
;
∣
G
:
H
∣
G; |G : H |
G
;
∣
G
:
H
∣
the index of
H
H
H
in
G
;
∣
X
∣
G; |X |
G
;
∣
X
∣
the number of elements of
X
⊆
G
;
Z
(
G
)
X \subseteq G; Z (G)
X
⊆
G
;
Z
(
G
)
the center of
G
;
G
′
G; G'
G
;
G
′
the commutator subgroup of
G
;
N
G
(
H
)
G; N_{G}(H )
G
;
N
G
(
H
)
the normalizer of
H
H
H
in
G
;
C
G
(
H
)
G; C_{G}(H )
G
;
C
G
(
H
)
the centralizer of
H
H
H
in
G
G
G
; and
S
n
S_{n}
S
n
the
n
n
n
-th symmetric group. Suppose
k
≥
2
k \geq 2
k
≥
2
is an integer such that for all
x
,
y
∈
G
x, y \in G
x
,
y
∈
G
and
i
∈
{
k
−
1
,
k
,
k
+
1
}
i \in \{k-1, k, k+1\}
i
∈
{
k
−
1
,
k
,
k
+
1
}
the relation
(
x
y
)
i
=
x
i
y
i
(xy)^{i}= x^{i}y^{i}
(
x
y
)
i
=
x
i
y
i
holds. Show that
G
G
G
is Abelian.
2
2
Hide problems
Determine all polynomials such that f(x^2-2x)=f^2(x-2)
Determine all polynomials
f
(
x
)
f (x)
f
(
x
)
with real coeffcients that satisfy
f
(
x
2
−
2
x
)
=
f
2
(
x
−
2
)
f (x^{2}-2x) = f^{2}(x-2)
f
(
x
2
−
2
x
)
=
f
2
(
x
−
2
)
for all
x
.
x.
x
.
group with x\to x^n is an isomorphism
In the questions below:
G
G
G
is a finite group;
H
≤
G
H \leq G
H
≤
G
a subgroup of
G
;
∣
G
:
H
∣
G; |G : H |
G
;
∣
G
:
H
∣
the index of
H
H
H
in
G
;
∣
X
∣
G; |X |
G
;
∣
X
∣
the number of elements of
X
⊆
G
;
Z
(
G
)
X \subseteq G; Z (G)
X
⊆
G
;
Z
(
G
)
the center of
G
;
G
′
G; G'
G
;
G
′
the commutator subgroup of
G
;
N
G
(
H
)
G; N_{G}(H )
G
;
N
G
(
H
)
the normalizer of
H
H
H
in
G
;
C
G
(
H
)
G; C_{G}(H )
G
;
C
G
(
H
)
the centralizer of
H
H
H
in
G
G
G
; and
S
n
S_{n}
S
n
the
n
n
n
-th symmetric group. Suppose that
n
≥
1
n \geq 1
n
≥
1
is such that the mapping
x
↦
x
n
x \mapsto x^{n}
x
↦
x
n
from
G
G
G
to itself is an isomorphism. Prove that for each
a
∈
G
,
a
n
−
1
∈
Z
(
G
)
.
a \in G, a^{n-1}\in Z (G).
a
∈
G
,
a
n
−
1
∈
Z
(
G
)
.